
Book ^fLSL 



THE COLLECTED 
^ MATHEMATICAL PAPERS 

OF 
JAMES JOSEPH SYLVESTER 

F.R.S., D.C.L., LL.D., Sc.D., 

Honorary Fellow of St John's College, Cambridge; 

Sometime Professor at University College, London ; at the University of Virginia ; 

at the Royal Military Academy, Woolwich ; at the Johns Hopkins University, Baltimore 

and Savilian Professor in the University of Oxford 



VOLUME I 



(1837-1853) 



Cambridge 

At the University Press 
1904 



QA3 



CTambTttise : 

PRINTED BY J. AND C. F. CLAY, 
AT THE UNIVERSITY PRESS. 

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MATHEMATICAL PAPERS 



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[A// Rigl.ts reserve<I\ 



PREFATORY NOTE. 

riiHE object aimed at in this volume has been to present a faithful record 
of the course of the author's thought, without such additions as recent 
developments of the subjects treated of might have afforded, and without any 
alterations other than that considerable number involved in the attempt to 
make the algebraical symbols read as the writer intended. While, for the 
reader's convenience, the author's references to his own papers have been 
accompanied by cross references to the pages of this volume, placed in square 
brackets. 

By far the longest paper in the volume is No. 57, " On the Theory of the 
Syzygetic Relations of two Rational Integral Functions, comprising an 
application to the Theory of Sturm's Functions," and to this many of the 
shorter papers in the volume are contributory. 

The volume contains also Sylvester's dialytic method of elimination 
(No. 9, etc.), his Essay on Canonical Forms (No. 34), and early investigations 
in the Theory of Invariants (Nos. 42, 43, etc.). 

It contains also celebrated theorems as to Determinants (Nos. 37, 39 48, 
etc.) and investigations as to the Transformation of Quadratic Forms (the 
Law of Inertia, No. 47, and the recognition of the Invariant factors of a 
matrix, Nos. 22, 24, 36). 

A full table of contents is prefixed. 



H. F. BAKER. 



St John's College, Cambridge. 
April, 1904. 



TABLE OF CONTENTS 

PAGES 

1. Analytical development of FresneVs optical 

theory of crystals . . . . . 1 — 27 

(Philosophical Magazine 1837, 1838) 

2. On the motion and rest of fluids . . 28 — 32 

(Philosophical Magazine 1838) 

3. 071 the motion and rest of rigid bodies . 33 — 35 

(Philosophical Magazine 1839) 

4. On definite double integration, supplementary 

to a former paper on the motion and rest 

of fluids 36—38 

(Philosophical Magazine 1839) 

6. On an extension of Sir John Wilson's theorem 

to all numbers whatever .... 39 

(Philosophical Magazine 1838) 

6. Note to the foregoing ..... 39 

(Philosophical Magazine 1839) 

7. On rational derivation from equations of 

coexistence, that is to say, a new and 

extended theory of elimination, Part I. 40 — 46 

(Philosophical Magazine 1839) 

8. On derivation of coexistence. Part II., being 

the theory of simidtaneous simple homo- 
geneous equations 47 — 53 

(Philosophical Magazine 1840) 

9. A method of determining by mere inspection 

the derivatives from two equations of 

any degree 54 — 57 

(Philosophical Magazine 1840) 

10. Note on elimination ..... 58 

(Philosophical Magazine 1840) 



CONTENTS. vii 

PAGES 

1 1 . On the relation of Sturm's auxiliary functions 

to the roots of an algebraic equation . 59, 60 

(Plymouth British Association Report 1841) 

12. Examples of the dialytic method of elimina- 

tion as applied to ternary systems of 

equations 61 — 65 

(Cambridge Mathematical Journal 1841) 

13. Introduction to an essay on the amount and 

distribution of the midtiplicity of the 

roots of an algebraic equation . . 66 — 68 

(Philosophical Magazine 1841) 

14. A new and more general theory of multiple 

roots 69—74 

(Philosophical Magazine 1841) 

15. On a linear method of eliminating between 

doid>le, treble, and other systems of 

algebraic equations 75 — 85 

(Philosophical Magazine 1841) 

16. Memoir on the dialytic method of elimina- 

tion, Fart I. 86—90 

(Philosophical Magazine 1842) 

17. Elementary researches in the analysis of 

combinatorial aggregation . . . 91 — 102 

(Philosophical Magazine 1844) 

18. On the existence of absolute criteria for de- 

termining the roots of numerical equations 103 — 106 

(Philosophical Magazine 1844) 

19. An account of a discovery in the theory of 

numbers relative to the equation 

Ax' + By' + Cz' = Dxyz . . 107—109 

(Philosophical Magazine 1847) 

20. On the equation in numbers 

Ax'-\- By'' + Cz" — Dxyz, 
and its associate system of equations . 110 — 113 

(Philosophical Magazine 1847) 

21. On the general solution, in certain cases, of 

the equation x^ + y^ + z^ — Mxyz . . 114 — 118 

(Philosophical Magazine 1847) 



viii CONTENTS. 

PAGES 

22. On the intersections, contacts, and other cor- 

relations of two conies expressed hy 
indeterminate coordinates . . . 119 — 137 

(Cambridge and Dublin Mathematical Journal 1850) 

23. An instantaneous demonstration of PascaTs 

theorem by the method of indeterminate 
coordinates 138 

(Philosophical Magazine 1850) 

24. On a new class of theorems in elimination 

between quadratic functions . . . 139 — 144 

(Philosophical Magazine 1850) 

25. Additions to the articles ' On a new class of 

theorems', and ' On Pascal's theorem,' . 145 — 151 

(Philosophical Magazine 1850) 

26. On the solution of a system of equations in 

which three homogeneous quadratic func- 
tions of three unknown quantities are 
respectively equated to numerical midtijiles 
of a fourth non-homogeneous function of 
the same 152 — 154 

(Philosophical Magazine 1850) 

27 . On aporismatic property of two conies having 

with one another a contact of the third 

order ....... 155, li56 

(Philosophical Magazine 1850) 

28. On the rotation of a rigid body about a fixed 

point ....... 157 — 161 

(Philosophical Magazine 1850) 

29. On the intersections of two conies . . 162 — 164 

(Cambridge and Dublin Mathematical Journal 1851) 

30. On certain general properties of homogeneous 

functions ...... 165 — 180 

(Cambridge and Dublin Mathematical Journal 1851) 

31. Reply to Professor Boole's observations on 

a theorem contained in last November 

number of this Journal. . . . 181 — 183 

(Cambridge and Dublin Mathematical Journal 1851) 



32. Sketch of a memoir on elimination, trans- 

formation and canonical forms . . 184 — 197 

(Cambridge and Dublin Mathematical Journal 1851) 

33. On the general theory of associated alge- 

braical forms 198 — 202 

(Cambridge and Dublin Mathematical Journal 1851) 

34. An essay on canonical forms, supplement to 

a sketch of a memoir on elimination, 
transformation and canonical forms . 203 — 216 

(George Bell, Fleet Street, 1851) 

35. Explanation of the coiiicidence of a theorem 

given by Mr Sylvester in the December 
number of this Journal with one stated 
by Professor DonMn hi the June number 
of the same 217, 218 

(Philosophical Magazine 1851) 

36. An enumeration of the contacts of lines and 

surfaces of the second order . . . 219 — 240 

(Philosophical Magazine 1851) 

37. On the relation between the minor deter- 

minants of lineai'ly equivalent quadratic 

functions 241 — 250 

(Philosophical Magazine 1851) [See p. 647 below.] 

38. Note on quadratic functions and hyper- 

determinants 251 

(Philosophical Magazine 1851) 

39. On a certain fundamental theorem of de- 

terminants 252 — 255 

(Philosophical Magazine 1851) 

40. Extensions of the dialytic method of elimina- 

tion 256—264 

(Philosophical Magazine 1851) 

41. On a remarkable discovery in the theory of 

canonical forms and of hyper determinants 265 — 283 

(Philosophical Magazine 1851) 

42. On the princip)les of the calcidus of forms . 284 — 327 

(Cambridge and Dublin Mathematical Journal 1852) 

43. On the principles of the calculus of forms . 328 — 363 

(Cambridge and Dublin Mathematical Journal 1852) 



44. Su7^ line propriete nouvelle de V equation qui 

sert a determiner les inegcdites seeulaires 

des planetes ...... 364 — 366 

(Nouvelles Annales de Mathematiques 1852) 

45. On a remarkable theorem in the theory of 

equal roots and multiple points . . 367 — 369 

(Philosophical Magazine 1852) 

46. Observations on a new theory of multiplicity 370 — 377 

(Philosophical Magazine 1852) 

47. A demonstration of the theorem that every 

homogeneous quadratic polynomial is 
reducible by real orthogonal substitutions 
to tJie form of a sum of positive and 
negative squares 378 — 381 

(Philosophical Magazine 1852) 

48. On Staudt's theorems concerning the contents 

of polygons and polyhedrons, with a 
note on a new and resembling class of 
theorems ■ 382 — 391 

(Philosophical Magazine 1852) 

49. On a sim])le geometrical problem, illustrating 

a conjectured principle in the theory of ^ 

geometrical method .... 392 — 395 

(Philosophical Magazine 1852) 

50. On the expression of the quotients which 

appear in the application of Sturm's 
method to the discovery of the real roots 
of an equation 396 — 398 

(Hull British Association Eeport 1853) 

51. On a theorem concerning the combination 

of determinants 399 — 401 

(Cambridge and Dublin Mathematical Journal 1853) 

52. Note on the calculus of forms . . . 402, 403 

(Cambridge and Dublin Mathematical Journal 1853) 

53. On the relation between the volume of a 

tetrahedron and the product of the six- 
teen algebraical values of its superficies 404 — 410 

(Cambridge and Dublin Mathematical Journal 1853) 



\1 



54. On the calem 

of invaria 

(Cambrii 

55. Theoreme sur 

des equatio 

(Nouvfa 

56. Noiivelle metliode pow, ui . 

superieure et une limite inferieure des 
racines reelles d^une equation algebrique 
quelconque 424 — 428 

(Nouvelles Annales de Math^matiques 1853) 

57. On a theory of the syzygetic relations of two 

rational integral functions, comprising 
an application to the theory of Sturm's 
functions, and that of the greatest alge- 
hraical common measure . . . 429 — 586 

(Philosophical Transactions of the Eoyal Society of London 1853) 

58. On the conditions necessary and sufficient to 

be satisfied in order that a function of 
any number of variables may be linearly 
equivalent to a function of any less 
number of variables .... 587 — 594 

(Philosophical Magazine 1853) 

59. On Mr Cayley's impromptu demonstration 

of the rule for determining at sight the ^ 

degree of any symmetrical function of 

the roots of an equation expressed in 

terms of the coefficients .... 595 — 598 

(Philosophical Magazine 1853) 

60. A proof that all the invariants to a cubic 

ternary form are rational functions of 
Aronhold's invariants and of a cognate 
theoremfi for biquadratic binary forms . 599 — 608 

(Philosophical Magazine 1853) 

61. On a remarTcable modification of Sturm! s 

theorem 609—619 

(Philosophical Magazine 1853) 



rm s 
iding 
roots 



620—626 



lOr and 
-, ^... roots of any 
algebraical equation .... 627 — 629 

(Philosophical Magazine 1853) 

64. Note on the new rule of limits . . . 630 — 633 

(Philosophical Magazine 1853) 

65. The algebraical theory of the secular in- 

equality determinantive equation gene- 
ralised 634—636 

(Philosophical Magazine 1853) 

66. On the explicit values of Sturm's quotients 637 — 640 

(Philosophical Magazine 1853) 

67. On a fundamental rule hi the algorithm of 

continued fractions 641 — 644 

(Philosophical Magazine 1853) 

68. On a generalisation of the Lagrangian 

theorem of interpolation . . . 645, 646 

(Philosophical Magazine 1853) 

Note on Sylvester's theorems on determinants 

IN this volttme 647 — C50 



ANALYTICAL DEVELOPMENT OF FRESNEL'S OPTICAL 
THEORY OF CRYSTALS. 

[Philosophical Magazine, XL (1837), pp. 461—469, 537 — 541 ; 
xii. (1838), pp. 73—83, 341—345.] 

Thk following is, I believe, the first successful attempt to obtain the 
full development of Fresnel's Theory of Crystals by direct geometrical 
methods. Hitherto little has been done beyond finding and investigating 
the properties of the wave surface, a subject certainly curious and interesting, 
but not of chief importance for ordinary practical purposes. Mr Kelland, 
in a most valuable contribution to the Cambridge Philosophical Transactions*, 
has incidentally obtained the difference of the squares of the velocities of a 
plane front in terms of the angles made by it with the optic axes. I have 
obtained each of the velocities separately, and in a form precisely the same 
for biaxal as for uniaxal crystals. 

I have also assigned in my last proposition the place of the lines of 
vibration in terms of the like quantities, and that in a shape remarkably 
convenient for determining the plane of polarization when the ray is given. 
For at first sight there appears to be some ambiguity in selecting which of 
the two lines of vibration is to be chosen when the front is known. If p be 
the perpendicular from the centre of the surface of elasticity let fall upon 
the front, ti, i,^ the angles made by the front with the optic planes, ei, 62 the 
angles between its due line of vibration and the optic axes, I have shown that 

//b'^~p^ sin tA //b^ — p^ sint2^ 

cos ei = . / — — i-- . -. — - , cos 6, = . / , , . -. — ' , 
V Va^ - c^ sm ij "V W - & sm ij ' 

so that all doubt is completely removed. The equation preparatorj' to 
obtaining the wave surface is found in Prop. 6 by common algebra, without 
any use of the properties of maxima and minima, and various other curious 
relations are discussed. 

Without the most careful attention to preserve pure symmetry, the 
expressions could never have been reduced to their present simple forms. 

* See 1,0116,. and, Edinb. Phil. Mag. Vol. x. p. 336. 



2 Analytical Development of [1 

Analytical Reduction of Feesnel's Optical Theoey of Crystals. 

Index of Contents. 

In Proposition 1, a plane front within a crystal beiag given, the two lines 
of vibration are investigated. 

In Proposition 2 it is shovm that the product of the cosines of the iacHna- 
tions of one of the axes of elasticity to the two lines of vibration, is to the 
same for either other axis of elasticity ia a constant ratio for the same crystal; 
and the two liaes of vibration are proved to be perpendicular to each other. 

In Proposition 3, a line of vibration being given, the front to which it 
belongs is determined; and it is proved that there is only one such, and 
consequently any line of vibration has but one other Hne conjugate to it. 

In Proposition 4, certain relations are instituted between the positions of, 
and velocities due to, conjugate lines. 

In Proposition 5, the angles made by the front with the planes of 
elasticity are found in terms of the velocities only. 

In Proposition 6, the above is reversed. 

In Proposition 7, the position of the planes in which the two velocities 
are equal (viz. the optic planes) is determined. 

In Proposition 8, the position of a front in respect to the optic axes is 
expressed in terms of the velocities. 

In Proposition 9, the problem is reversed, and it is shown that if v^, v, 
be the two normal velocities with which any front can move perpendicular 
to itself, and ij, i„ the angles which it makes with the optic planes, then 



v^ = a- ( si 



sm'L^-j+c' cos^'^ 



%- = a- ( sm — — ^ 1 + & I cos — „- 



In the 10th the angle made by a line of vibration with the axes ot 
elasticity is expressed in terms of the two velocities of the front to which it 
belongs. 

In the 11th Proposition the velocity due to any line of vibration is ex- 
pressed in terms of the angles which it makes with the optic axes, viz. 

'if- — h° = (a- — c") cos 6i cos 62. 
In the 12th Proposition Cj, 62 are separately expressed in terms of ti, ta- 
in the Appendix I have given the polar or rather radio-angular equation 

to the wave surface, from which the celebrated proposition of the ray flows as 

an immediate consequence. 



IJ FresneVs Optical Theory of Crystals. 



Proposition 1. 

If lx + my + 7iz = (a) 

be the equation to a given front, to determine the lines of vibration therein. 

It is clear that ii x, y, z be any point in one of these lines, the force 
acting on a particle placed there when resolved into the plane must tend to 
the centre. Consequently the line of force at x, y, z must meet the perpen- 
dicular drawn upon the front from the origin. Now the equation to this 
perpendicular is 

I m. n ' 

and the forces acting at x, y, z are arx, b^y, c-z parallel to x, y, z, so that the 
equation to the line of force is 

X-.^Y-y^Z-^_ ■ 

a-x b-y &z 

From (2) we obtain 

b'-yX — a-x Y =(b- — a^) xy (3) 

c-zY - b'-yZ = (c= - b^) yz (4) 

a^xZ — c^zX = (a^ — c^) zx. (5) 

Hence 

(&^ — a^) xyn + (c^ — 6^) yzl + (a^ — c'^) zxm 

= % {nX - IZ) + (fz (lY- mX) + a'x (mZ- 7iY) ; 
but by equations (1) 

lZ-nX = 0, mX-lY=0, nY-mZ=0 
therefore 

(&'- -a')- + (c' -b^)- + (a= - cO - = 0. (6) 

' z ^ ' X ' y 

Also we have 

nz -\-lx + my = (a) 

therefore 

(6= - a^) n^ + (c= - &2) P + nl Uc- - 6^ - + (&' - «') 7) = (»' - c') »i' 
or 

(c^ - ^) (3' + ^ [{C^ - h^ Z= + (6= - a=) n^ - (a= - c-) «i=} ^ + (6^ - «0 = 0. 
And in like manner interchanging b, y, m with c, z, n 

(b". _ c') gy + _!. f(6= _ c=) Z^ + (c= - a^) «,= - (a^ - 60 «^} f + (c= - a^) = 0. 

1—2 



A7ialytical Development of [1 

Hence if (^, — ) (— , — ) be the two systems of values of -, -, then 



\K x^' X Xi' \X x„ ' X xj 
are the two lines of vibration required. 

Proposition 2. 
By last proposition it appears that 



x,x. 




h' 


-& 


z^z^ 




6= 


~a^ 


x,x. 




c^ 


-Jf 


2/12/2 + ^■1^2 




c^ 


-b"- 



(c) 



and ~- = -^zi:^ (d) 

therefore _ 

aJiflj, b- — c- 

therefore 

^1*2 + 2/12/2 + •^1^2 = 0- 

And therefore the two lines of vibration are perpendicular to each other. 
N.B. Equations (c) and (d) must not be overlooked. 

Proposition 3. 

A line of vibration is given ( that is ~ , — are given ] and the position of 
the front is to be determined. 

Let Ix + my + nz=0 be the front required, then Ix^ + my^ + nz^ = 0, and 

(6=-cO-+(c=-a=)- + (a— 6=)- = 0. 
«i 2/i «i 

Eliminating 7i we get 

I ((a^ - ¥) I - (6^ - cO I) + m ((a^ - &0 | " (0^ - o?) |) = 

therefore 

I _Xi (a^ — b") y^ — (c^ — a^) z^ 
m 2/1 (6^ — &) z^ — (a^ — 6^) ajj^ 

_ X-, a- (.«i^ + 3/1^ + giO - (ctV + 6 V + cV) 

~ yi &' («i' + 2/1' + ^i) - ("'W + b'yi + cW) ' 



1] FresneVs Optical Theory of Crystals. 5 

If now we make x^ + y^- + ^f = 1 

a?x^- + }pyx~ + c^z^ = v-^ 

and therefore 

I _Xi a^ — ■yi" 
m y^' b^ — v{^ 

and in like manner 

Z _ a?! a- — Vi _ 
m 2^1 " c" — Vi^ ' 

therefore 

(a^ — «!-) «!« + {h"- — vf) y^y + (c' — ■Wj^) ^i^ = 

is the equation required. 

Proposition 4. 

— , - having each only one value, shows that only one front corresponds 

to the given line of vibration. Let x^, y^, z^, v^ correspond to (c^, y^, z^, v^ for 
the conjugate line of vibration, then the equation to the front may be 
expressed likewise by 

(a^ — v/) x^ + Qf — Vo') y^y + (c^ — v^') z^^z = 0, 

so that 

(g^ - v^^) x^ ^ Qf - Vi"^) yi ^ (c^ - vj") z^ 
{o? - vi) «2 Qf - vi) 2/2 (& - vi) z^ ' 

Proposition 5. 

To find CO, ^, ■xjr, the angles made by the front with the 'planes of elasticity 
in terms of v-ijV^. 

By the last proposition 

, „ {a--viYxi 

(cos ft)/ = '^—, ^;r^ — ., , ,7, :;^ „ , , , ^„ — 7. 

^ ' {a"" - v^-y X:c + {b- - V-:)- y-c + {c- - viy z^- 

__ (g^ — vf) (a- — vi) x-^x^ 

~ (a^ - vi) (a^ - vi) x^x^ + (6- - vi) (b"- - vi) yiy._ + (c' - vi) (c- - vi) z^z^ ' 

Now, by Proposition 2, 

a^A ^ yxyi ^ z^z^ 
c^ — b- a^ — & ¥ — a^ 



6 Analytical Development of [1 

therefore (cos &>)- 

{a'-v^''){a''-vi){c--¥) 

~ {d'-v;') {a'-vJ'){c'-b')+{b' - v^'){b' - vi){a? - &) + (c- - v,^) {c^-v^'){b' - a') 

(a- - Vj') (g^ - Vj-) (c- - b-) 

~ a' {c- - 6^ + 6" (a- - c=) + c' (a- - b") 

Similarly, 






Proposition 6. 

To find Vi, V.2 in terms of co, <f), •\|r. 

By the last proposition 
(cos my 



therefore 



a" - ^1= (a= - 60 («' - c") ' ■ (a' - b'} (a^ - c^ 

(cos (^)- 6^ 2 1 

b^-v,' ~ (6- - a'O {b"- - c=) "'' ■ {¥ - a') {¥ - c") 

(cos -v^)- _ C" _ ., 1 

c- - Vi' ~ (c^ - b') (c- - a'O ''''■(c-^-a-)(c'-6') 

(cos o))'- (cos (/>)^ (cos ■y^y _ 
a'' — 1)1^ ¥ — v^ & — v^ 



Just in the same way 

(cos aif (cos <^y (cos •^)- _ . 
ci^ _ vo^ '^ b^- vi '^ C-- v^^ ~ ' 

so that Vi^, V2 are the two roots of the equation 

(cos of (cos <^)- (cos -v^)'- 

— ; r "I — TT, r H ; 5~ — "• 

a- — V-' 0- — ■«- c- — w- 

COR. Hence the equation to the wave surface may be obtained by making 

(cos co)x + (cos 4>)y + (cos ■y^)z = v, 



1] FresneTs Optical Theory of Crystals. 

or if we please to apply Prop. 5, we may make 






or, if we please *, 

/( aV-l)(a'-^0 /(6V-1)(6^-^) 

/{chi? - 1) {d" - ■wO 



■/i 



.2 = 1. 



id) 



{c^ - a') (c^ - h") 

Proposition 7. 
To find when Vi = v^. 

By Prop. 4, 

Hence whea Vi = v.2 we have, generally speaking, 

X-l 2/2 Z2 

Now a;ia;2 + 2/12/2 + ^1^2 = ; 

therefore x^ + j/,^ + zf would = 0, which is absurd. 

The only case therefore when v-^ can = ii^ is when one of those terms of 

a;, ^ 

equation {&) becomes ^ : thus suppose v-^ = h, then we have ~ = ~ = k > ^'^^ 

we can no longer infer — = — . 
x„ 2/2 

Let now (coi, ^1, -\|rj)((U2, <^^, ■<if^ be the two systems of values which oj, (^, 
•\|r assume when Wi = ■«„ = &, then applying the equation of Prop. 5 we have 



la?—h' 

cos Ui^=^ . —: r 


/a? 


-h-" 


cos 0,2 ^ ^, 


-& 


cos </>i = 


cos ^2 = 




, /6^ - c'' 


cos^ /^' 


-& 


COSt2-^^2 


-& 


so that h must correspond to the mean 


axis. 




[* See below, 


p. 27. Ed.] 





8 Analytical Development of [1 

Proposition 8. 

ti, t2 heing the angles made by the front with the optic planes, to find 
ti, t2 in terms of Vi, v^. 

By analytical geometry 

cos li = cos ft) . cos (Uj + cos (p . cos ^1 + cos yjr . cos i|ri 

_ / (vi" - a-) {vj - g") / a^-6^ 

/ (t;i' - C- ) (z;^" - cO l&-\P^ 
V (c^ - a=) (c^ - 6^) ■ V c' - a- 

" a'-c^ 

and similarly 

cos (2 = cos ft) . cos (1)2 + cos cf> . cos <f)o + cos -v/r . cos ^fr.^ 

V{(di^ - g^) (v^- - a°)} - V{(^i' - cQ (^2- - cQ} 
a- — c- 

Proposition 9. 

To find Vi, v^ in terms of h, u. 
By the last proposition 

cos ., . COS . = fa--«-)fa--;-)-fa--c-)W-c°-) 

^ {a'-c')-{a-'-&){v^ + v;-) 

_ (g- + c') - (^Ji" + vj) 
{a^ - c^) 
therefore 

v-^ + vi = g^ + c^ — (g^ — c^) cos tj cos i^. 

Again, (sin i-^f . (sin ^2)^ = 1 — (cos ti)^ — (cos tj)^ + (cos i-^^ (cos ta)" 



+ 



(g= - &f 
(g^ + c^f - 2 (g^ + c°-) (w^^ + v^) + («i= + ^aO' 



(g= - c=)= 
(g= - c^Y 



1] 



FresneTs Optical Theory of Crystals. 



therefore 

but 
therefore 



t>i^ — vi = {or — c") sin ti . sin tj 
Vi + •Va^ = (a^ + C") — (a- — c'-) cos ti cos i^ 



a' + c^ a-— c- , , 

Wi = ?;^ ?; COS (ti + lo) 



= a-' Sin -, 



2 
ti 4- toA" 



+ C^ COS 



*1 + '2\' 



COS (ti — ta) 



a" + c" a- 



= a=(sin^)Vc^(cosVT. 

Thus for uniaxal crystals where tj + ta = 180° 
v^ = a^ 
v^ = a? (cos t)^ + c^ (sin i)l 

Cor. Hence we may reduce the discovery of the two fronts into which 
a plane front is refracted on enter- 
ing a crystal to the following trigo- 
nometrical problem. 

Let a sphere be described about 
any point in the line in which the 
air front intersects the plane of in- 
cidence. Let the great circle PI 
denote the latter plane, IF the 
former, OA, OC also great circles, 

the planes of single velocity. Sup- Fig. i 

pose IGH to be one of the refracted 
fronts intersecting OA, OC in and H, then 

(a- + c')- (a'' -c') cos (G + H)_ (sin PIFf 




2 (vel. in air)= (sin PIGHy " 

The double sign will give rise to two positions of the refracted front IGH. 

The propositions which follow are perhaps more curious than immediately 
useful. 



10 Aiialytical Development of [1 



Proposition 10. 

To determine the portion of a line of vibration in terms of the two 
velocities of its corresponding front. 

We have here to determine the quantities — , — (of Prop. 1) in terms of 
Vi, v^,OT on putting a;i^ + 3/1^ + ^1^=1, x^ , y^ , z^ are to be found in terms of Vi, v^. 

By Prop. 3 

I m n 

Xi-.yi-.Zi- 



' a= - vi' ' b^-Vi"' (f- Vj' 

: m^ : 71^ : : (b- — c^) (a- — v-c) (a^ — V2') 
: (c- - a") {¥ - Vi") (b^ - vi) 

a- — v^ b^ — Vi' c^ — Vi^ 

Let a, /3, 7 be the angles made by the given line of vibration with the 
elastic axes, then 

™ 2 
(cos a)^ = 



and by Prop. 5 



therefore 



= (b"- - c") {a? - v.?) (6- - v^) (c= - z/iO 
divided by 

(6- - C-) (a" - Vo^) (b- - v^) {c- - Vi^) + (c- - a-) (6= - v/) (c- - 1;/) (a^ - vf) 

+ (a- — 6^) (c- — v.^ {a- — ■Wi^) (b- — Vi) 
and therefore 

_ (i!)^ - c^) (g^ - vj) (6' - zii") (c' - •j^i") 

~ (Vi^ - ■ya^) (a- - 6^) {¥ - c^) {(!' - a') 

(where it is to be observed that the reduction of the denominator is simply 
the effect of a vast heap of terms disappearing under the influence of 
contact with the magic circuit (a" — b^), (b- — c'), (c- — a'), a simpler instance 
of which was seen in Proposition 5). 

In fact the coefficient of v* . v"- 

= (&- - C-) + (c= - a") + (a^ - b^) 

= 



1] FresneTs Optical Theory of Crystals. 11 

that of v^'' . v«} = (c- + 6-) . (c^ - Jf) 

+ (a- + cO . (a^ - C) 
+ {b^ + a') . (6= - a") 

= 0. 

The term in which neither Vi nor v« enters 

= a%-c' {{b" - C-) + (c' - aO + (a- - &^)) 
= 0. 
The coefficient of 

- vi' = aP.{¥- c^) + b^ . (c' - a') + c" . (a* - b*) 
and that of 

v/ = b^c^ . (c- - b-) + c'a^ . (a= - c') + a%^ . (b^ - a") 
each of which 

= {a''-b-).{b"--c^).(c'-a'). 
Hence 

^ ^ Vi'- vi ' (a^ - b^) (a- - c") ' 

in like manner (cos /3)^ = &c. 

, / V «i'-6' (c'-vi)(a''-vi) 
and (cos y)- = . ^, ff4 -r . 

Proposition 11. 

6i, 62 being the angles between any Line of vibration and the optic axes, 
required the velocity due to that line in terms of ej, €3. 

By analytical geometry, 

cos 61 = cos a . cos ^1 + cos 7 . cos i^r^ 
cos 62 = cos a. . cos ^1 — cos <y . cos iIti 
therefore cos ei . cos e^ = (cos a)^ (cos c/),)^ — (cos 7)^ (cos 'y^if 

Vj^ - 6^ Ua^ - vi) . (c^ - vi) - (c^ - vi) . (a"- - Vi")) 



vi -vi' { (a^ - c^y 

^ vi - b^ (a^-c^){vi-vi ) 
vi-vi' (a^-c^f 

_b^ — vi 
a^ — c^ ' 

Hence vi = 6^ — (a^ — c^) cos 61 cos e^, 

and in like manner, for the conjugate line of vibration 

vi = b' — (a- — C-) cos e/ cos 63'. 



12 Analytical Development of [1 

Proposition 12. 
To find €i, 62 in terms of ti, ta- 

(cos 6])^ + (cos 62)^ = 2 (cos a)= , (cos (/)i)= + 2 (cos 7)- . (cos 1/^1)- 

== 2 ^^'~^' [ (g^ - i;/) . (c° - 2;i^) + (c^ - •i)^'') ■ (cr - v^) ] 
v^^ - vi \ (cC- - dj J 

but by Prop. 9 

v-^ = a- I sm — - — ) + c- ( cos — ^ — 1 

V = a" I sin — g— 1 + c- (cos — 2" 
therefore 



(cos 61)- + (cos 62)- = -—^ 



{a? — C-) sin c^ . sin i^ 
multiplied by 

2 (a^ - c^^ [(cos ^j^ (sin 'i^j^ + (cos ^)^ (sin '^^^ 



(a= - c'f 
b" - Vi' 



{a^ — c^) sin tj . sin i^ 

and we have seen that 

b- - V'- 



{(sin iif + (sin lof 



cos fii cos 60 = 



therefore 



therefore 



cos fii + cos eo = 



a-"— c- 

/Z)- — tij-N sin ti + sin i 



cos e,- cos 62=^ (^,^,^2 j 



a^ — cV ' V(siiiti- sinta) 
6^ — ^i^N sin ti — sin I2 



V(sinti.sin|t2) 



sm t, 
sin tj 
and in like manner 

{b^ — v^ sin ti 
■ c^ " sin t. 



, / (6^ — v„^ sin ti 

cos e., = . / -^ „ '- . -. — , 

V {O' — c- sm tij 

where v^, v^ for the sake of neatness are left unexpressed in terms of ti, 



1] 



FresneVs Optical Theory of Crystals. 



13 



This is the simplest form by which the position of the lines of vibration 
can be denoted. 

COK. From the last proposition it appears that 

cos 61 _ sin ti 
cos 62 sin to " 

Hence we may construct geometrically for the two planes of polarization. 

Let /, K be the projections of 
the two optic axes on a sphere, E 
the projection of the normal to the 
front, P the projection of one line 
of vibration ; then 

cos PK _ sin KE 
cos PI " sin IE ' 

Draw FEG the circle of which 
P is the pole, meeting PK, PI pro- 
duced in G and F. 



Then 


cos PK= sin KG, 


\ 


and 


cos PI = sin IF, 


\ 


therefore 


sin KG sin KE 
sin FI " sin IE 


\ 


therefore 


-^. 






sin KG 


sin IF 




sin KE ~ 


' sin IE 




Fig. 2. 



therefore 

sin KEG = sin lEF 

therefore KEG=IEF or 180° -lEF. But PEF=PEG, therefore EP bisects 
either the angle lEK or the supplement to it. 

These two positions of EP give the two planes of polarization. The 
construction is the same as that given in Mr Airy's tracts, and originally 
proposed, I believe, by Mr MacCullagh. 



14 Analytical Development of [1 



ADDENDUM. 

If in the equation of Prop. 6, viz. 

(cos aif (cos 4>y^ (cos -^y _ 

a} _ j;2 ^2 _ ^2 g2 _ ^2 

we chancre a, b, c, v into - , y , - , - , and consider v to be the length 
" a b c V ^ 

of a line drawn perpendicular to the plane 

cos 0) .X + cos 4>.y + cos T|r . ^ = 0, 

the equation to the extremity thereof must be 

a^r^ (cos a>y b^r^ (cos <^)^ c^r^ (cos i/r)^ 
a^ — r- 6^ — r'^ c^ — r^ 

where w, ^, yjr denote the angles between the radius vector r, and the axes 
of X, y, z, so that the equation may be written 

a^a? y^y^ c^z^ 

+ iir^ 1 = 0, 



a? — r^ 6' 
which is that of the wave surface. 
But we have seen that 



v- = c- \ cos 



mv-H"^)]' 



therefore the equation to the wave surface may be written 



/ ti + i^ [ . ti + u 



where (j, i^ denote the angles between the radius vector v and the^two lines 

which would be the optic axes if a, b, c were chan£;ed into - , ^ , -so that 
^ ° a b c 

if e be the inclination of either to the meaa axis of elasticity 
'1 _ r 




_c //d-—b-\ 

^b\/w^') 



a // b"-c^ \ 
b\/ W-cV' 

These lines I shall call by way of distinction the prime radii*. 

* Upon the authority of Professor Airy I have appropriated the term optic axes to the Hnes 
normal to the fronts of single velocity. 



1] FresneVs Optical Theory of Crystals. 15 

be the two values of r corresf 



Cor. 1. If ^1, r^ be the two values of r corresponding to the same values 
of t] , ia we have 



; -. = --! I COS ^ 

1 

a- 
sin ti . sin tj, 



2 ) 



which proves the celebrated problem of two rays having a common direction 
in a crystal. 

Cor. 2. The intersection of any concentric sphere with the wave surface 
is found by making r constant. Hence Ly + tj becomes constant, and there- 
fore rii + rta = constant. Hence the curve of intersection is the locus of 
points, the sum or difference of whose distances from two poles when 
measured by the arcs of great circles is constant ; the poles being the points 
in which the prime radii pierce the sphere. 

In three cases these spherico-ellipses or spherico-hyperbolas become great 
circles : 

(1) When t] + t., = the angle between the two poles, in which case the 
curve of intersection is the great circle which comprises the two poles. 

(2) When l^ — u= 0, when the locus is a great circle perpendicular to 
the former and bisecting the angle between the optic axes. 

(3) When t, + 12= 180'', when the locus is a great circle perpendicular 
to the two above, and bisecting the supplemental angle between the two 
axes. 

Various other properties may be with the greatest simplicity deduced 
from the radio-angular equation. The hurry of the press leaves me time 
only to subjoin the following 

Proposition. 

To find the inclination of the radius vector to the tangent plane, in terms of 
the angles which the radius vector makes with the prime radii. 

Let be the centre of the wave surface, OA, OB the two prime radii, 
OP any radius vector. Let OP = v, POA = i^, FOB =1,2, and let the in- 
clination of the planes POA, POB = fi; 

then - = — — H — 



(taking only the positive sign for the sake of brevity). 



16 



Analytical Development of 



[1 



Let OQ, OR be the two adjacent radii vectores, so assumed that 

QOA = POA , QOB = POB + hu, 

ROB = FOB, ROA=POA+Bi„ 

and let p, q, r, a, b be the projections of P, Q, R, A, B on 
a sphere of which is the centre, then it is clear that 

qpa^ 90", rpb =90\ 

draw qm perpendicular to pb, then pm = 8u, and therefore 




_ pm _ pm _ 8t.j 
"' sin pqm sin a/)6 sin /i ' 



In like manner 



Now the angle QPO 



sin n 



also 



, _, r.POQ , . 



d.\ 



dr 
du, 



±L 



Bi, 



therefore 
therefore 



rrf., ^i'-'U -«.)«'" ^"-^''>- 



cot QPO = ^ (^ - Jj sin (i, + «,) sin fi. 
In like manner 

col KPO = — I Jsind, + <.)sina, 

4 Vc* av 



therefore 




QPO = RPO. 

Also it is clear that rpq = apb = fi. Ai '^ 
to find the inclinati.>n of OP to RPQ. wo ha ^^ 
only to describe a sphere of which P is tl'® 
centre, and intersecting PQ. PR, PO in ^- • 
/2'. 0'. 

Then iJ'O'Q' = /*. and 



Fig. 4. 



OQ' = O'R' = cot-' 



{i(^-a')''°^''+*'^''°^' 



1] FresneVs Optical Theory of Crystals. 17 

Draw O'N perpendicular to R'<^, then O'N measures the inclination of 
the radius vector to the tangent plane*. 

And qO'N = ^, 

therefore cos | = tan O'iV . cot O'Q', 

, . r..^^ cot O'Q' 

therefore cot iv = -^^— , 



and therefore 

Let A OB the angle between the optic axes = 2e, then by mere trigonometry 



t 



= ^'"'-(i-'-^)^'"^"-'''' 



cot O'N = i'- . (— - ^J sin I . sin ((, + 1^). 
between the optic axes = 2e, then b 
/sin (e + *' ., ^) sin (e- ' ., 
2 ~ V' ~in(,.sini-j 

itiun between tl 

,.(.+!^--)si„(.-"r-H) 



sm -„ = 



therefore tlie tangent of the inclination between the ra.iiiis vector and 
ihe normal 



sin (, .sin (. 

In like manner the tangent of the inclination between tho sj»iik- ratiius 
■ ^ctor and the normal at the other point of the wave-surface pierced by it 

" ,,J' M-, , /'K'-'^-)-'"('-n'-) 

a, =i('')'U'"ciJ''"^*'~*'*-\/ sin ...sin,. ' 

We may, in the .same way, find the inclination of the tfingLnt plane 

'■'< either of the prime radii, and to the plane which contains them both, 

0' terms of t, and «,; the former by a remarkably elegant construction ; 

,t the final expressions do not present themselves under the same simple 

pect. 

If we call 6 the angle between the ray and the front, we may still 

further reduce by substituting for r" its values in terms of «,, u, and we 

shall obtain 

•2 ( r- - r/ • ) 

cot d) = — 

'i I h . i 'i + 'j 

c' tan — g-- + u' cot 
X A /]■'''" (^ + '9 J "'" (e-*'-|-^].cosect, .cosec t,/. 



* O' is the projection of thi: rav and R'O' of the tangent plane. Therefore O'N being 
perpendicular to H'Q' represents their inclination. 

s. 2 



I 



18 Analytical Development of [1 

And if TTi, TTg be the inclinations of the normal to the two prime radii, 
it may be shown that 

cos TTi = cos (^ sin l^ + sin cos tj sin ^ , 

cos TTo = cos (^ sin i^ ± sin (^ cos t^ sin ^ . 

Cor. 1. For uniaxal crystals ^ = 90° and t, + t2=180°, so that the 
tangent of the inclination of normal to radius vector 

= Ir'^. ( — — A sin 2t for one point, 

and = for the other. 

Cor. 2. For every point in the circular section which passes through 
the poles sin „ = 0, and for the other two circular sections tj + t2 = or 180". 

Therefore every point in the three circular sections is an apse. 

Cor. 3. When a nearly = c, — ; is very small ; and therefore the 

normal and radius vector very nearly coincide. 

Cor. 4. Referring to fig. 4 we see that O'N bisects the angle Jt'O'Q'. 
Now R'O, Q'O are respectively perpendicular to the planes passing through 
0' and the optic axes ; and therefore the meridian plane as we may term it, 
that is, the plane containing both the ray and the normal, always bisects 
the angle formed by the two planes drawn through the ray and the two 
optic axes. 

Cor. 5. When 

li or to = 0, 

62 or ti = e. 

And therefore (f> assumes the form - , which indicates that the extremities 
of the four prime radii are singular points. 

In concluding for the present it behoves me to state that one step has 
been omitted in the foregoing paper*, viz. the actual performance of the 
eliminations which lead to the rectilinear equation to the wave-surface. 
But Mr Archibald Smith's elegant and brief Memoir in the Cambridge 
Philosophical Transactions'\ of last year leaves nothing to be desired further 
on that head. 

[* See below, p. 27. Ed.] [f Vol. vi. Also Phil. Mag. April, 1838, p. 335. Ed.] 



v 



IJ FresneVs Optical Theory of Crystals. 19 

That I have not exhibited it in its proper place (Prop. 6) arises only from 
my respect to the principle of literary propriety. With this important blank 
supplied the Analytical Theory may be pronounced to be complete. 

For all errors and imperfections in what precedes my excuse must be 
press of time and a total want of the materials to be derived from consulting 
works of reference. 



Since writing the above I have had an opportunity of reading the paper of our 
living Laplace inserted as part of the Third Supplement to his System of Eays in 
the Transactions of the Royal Irish Academy, in which the principal foregoing 
results are obtained by aid of a more refined and transcendental analysis. 

The nature of the four singular points is there discussed and the existence of 
four circles of plane contact demonstrated. 

The former may be very easily shown thus : when tj is very small i^=1e— i-^ cos \^ 
very nearly, ip denoting the inclination of the plane in which e is reckoned to the 
plane in which ij is reckoned. 



Hence 






therefore 



I - j^ x/{(«^ - b') (b^- - «^)} (cos ^ ± 1) .X, 



{'4< 



COSl/f+ 1) ( 1 



Take if/ constant and let the abscissae and ordinates be reckoned respectively along 
and perpendicular to the prime ray. 

Then 'i = - nearly, and r = J{y' + x") = x, 

or, if we change the origin to the other extremity of the prime ray. 



so that the equation becomes 



■ b — X, 



i(o»..i)y{(.-|)| 



20 



Analytical Development of 



[1 



Hence at each singular point the surface is touched by a cone, the equation to 
the generating line of which is given by the above, the extreme angle between it 
and the prime ray being 

»'-[y{M)(';-o}]- 



When h = a, ij/ always = ^ and the cone returns into a plane. 

Again, let us suppose that the position of any perpendicular from the centre 
is given, and that of the corresponding radius vector required. 

Let OA, OB* denote what we have termed the optic axes, but which it will be 
more agreeable to analogy to term the prime perpendiculars from centre, and let 
OP be the given normal. Take OQ, OR contiguous perpendiculars from centre 
in planes POQ, ROP, perpendicular to POA, POB respectively, then the inclination 
of the two former will be the same as that of the two latter, and may be 
termed fx. 

Let ii, 12 no'w denote the angles POA, POB respectively, then 
QOA=i^, QOB = u + hi^, 

ROA.= i, + Si„ ROB=u. 

The ray will be found by joining with the intersection of three planes drawn 
at P, Q, R, perpendicular to OP, OQ, OR, respectively. 

Now from Prop. 9 it appears that 



OP- 



Vf 



'1+^2^ 



(? ( cos - 



using only one sign for the sake of simplicity, which we may do by throwing the 
ambiguity upon the way in which i^ or i„ is measured, also 

0Q=0PJ-^''^ 




OR = OP + 



di., 
d.OP 



Let 811 = 812, then it is clear that OQ = OR, and 
the intersection of the two planes perpendicular to 
OQ, OR is therefore a line perpendicular to the 
plane QOR, and to the line which bisects the angle 
QOR. 

In fact if we draw QT, RT perpendicular to OQ, OR respectively in the plane 
QOR, the intersection in question passes through T and is perpendicular to 0^; 
also 

OT = OQ . sec (^ ROQ) = OQ 

to the first order of smallness. 



OA, OB are not expressed in the figure. 



1] FresneVs Optical Theory of Crystals. 21 



Now it 


is 


easy 


to 


see 


(just 


as 


on p. 16) that 


and also 














HOP 
QOP 


sin /a' 

sin /a' 



therefore EOF = QOP and therefore POT is perpendicular to QOR. 

Hence the problem is reduced to finding L the intersection of two lines TL, PL 
drawn in the same plane POT. 

Now because OTL, OPL are each right angles, a circle may be made to pass 
through L, T, P, 0. 

Hence the angle 

OP X POT 



PLO = P TO = tun-' 



OT-OP 



nn D/TD 1 OP X ■ ^ cos \fi. 

. -I OP X r'Oii . cos * 11, , _, sm u 

= *"" cl.OP, -^"" d.OP, ' 

j 01, z Ol„ 

ai2 " cCi2 

and OL = OP. sec POL. 

Also the position of the plane POL is known, and therefore the radius is 
completely determined in magnitude and position. 

It may be worth while also to remark that the above constructions enable us 
to form a series of equations between the magnitude of the radius and its incli- 
nations to the two prime perpendiculars. 

In fact, if we call tt^, tt.^ the two inclinations in question 

cos TTj = cos POL cos ij + sin POL sin tj . sin ^ , 

cos TTj = cos POL cos t2 + sin POL sin t.2 . sin -= , 

and of course if we call the angle between the two prime normals 2E 

.(^.i^--)sin(^-^^) 
sin ij sin ij 

Cor. 1. When i^ or 1^ = 0, tan POZ assumes the form ^ which may be 

interpreted analogously to the method used in the reverse problem, but may be 
more elegantly illustrated by 

Cor. 2. "Which is that the meridian plane POT (that is, the plane in which 
both normal and radius lie) bisects the angle formed by POP, QOP, and therefore 



Analytical Development of 



[1 




that formed by the planes drawn through the normal and the two prime normals 
to which these two are perpendicular. 

Now we have found (Cor. 4, page 18), that it also bisects the angle formed by 

the two planes passing through the 
radius and the two prime radii. Hence 
when the ray is given, we may find 
by the easiest geometry the normal 
and the tangent plane, and vice versa. 

Thus suppose (iT, N') {R, R) to be 
the projections of the prime perpen- 
diculars and prime radii on a sphere 
concentric with the wave surface. 

Let n be the projection of any 
given perpendicular on the same 
sphere ; join nN, nN' ; bisect NnN' by nM, which will be the meridian plane. 

Draw from R', R'TV perpendicular to nM and make B'T^TV. Produce RV 

to meet Mn in r, then RrM = R'rM. and 
therefore r is the projection of the radius. 
Just in the same way when r is given we 
may find n. 

Now suppose n to come to N, then 
the position of the meridian plane nM 
becomes indeterminate, and r from a point 

becomes a locus, subject to the condition that R'rN = RrN. From r draw rB 

perpendicular to rN. 

Then it is clear that because rN bisects RrR' 

sin RD _ sin Rr sin RJV 
sin 7i!'Z' sin.SV sin i^'iV"' 
and therefore Z> is a fixed point and JfD a fixed length, and 

cos rNB = tan rN . cot ND ; 

therefore the projection of the locus of r upon a plane drawn at N perpendicular 
to the line joining N with the centre is given by the equation 

p = ON . cot ND . cos 6, 

N being the origin and the projection of ND the prime radius ; which is the 
equation to a circle passing through N, and whose diameter = ON cot ND. 

Hence at the extremity of each prime perpendicular the tangent plane meets 
the surface in a circle passing through that extremity and whose radius = |6 cot a, 
a being to be found from the equation 

sin {2E 4. a) ^ sin {E + e) 




Fig. 7. 



that is 



sin a sin [E — e)' 

tan (E + a)= (tan Ef cot e. 



1] FresneVs Optical Theory of Crystals. 23 

Just in the same way it may be shown that the trace of the perpendiculars to 
the tangent planes of the surface at the point where it is pierced by any prime 
radius upon a plane perpendicular to that radius at its extremity, is also a circle 
passing through it, and curved in an opposite direction from the circle of plane 
contact nearest to it. 

Hence the enveloping cone at these points may be described as being perpen- 
dicular to the circular cone, formed by drawing lines from the centre to the above 
described circle ; that is every generating line of the one will be perpendicular to 
the generating line which it meets of the other. 

More generally it easily appears from fig. 6 that if a series of great circles 
(representing meridian planes) be taken intersecting the great circle NRR'N' 
in a fixed point, a plane perpendicular to the radius passing through that 
point, will intersect the cone of rays as well as the cone of perpendiculars corre- 
sponding to those meridian planes, in two circles. So that there exist an indefinite 
number of circular cones of rays corresponding to circular cones of perpendiculars 
touching each other in a line lying in the plane containing the extreme axes, and 
having their circular sections perpendicular to that line. 

The cusps are explained by the cone of rays degenerating into a right line, and 
the circles of plane contact by the cone of perpendiculars so degenerating. 

Furthermore I observe in conclusion that when a ray is given it follows from 
the general geometrical construction above that there will be two meridian planes 
according as we take R with R', or with a point 180° from R', and consequently 
these two planes will be perpendicular to each other. 

And similarly when a normal is given there will be two meridian planes per- 
pendicular to each other. 

Thus the planes passing through any radius and the two normals at the points 
where it pierces the wave surface, are perpendicular to each other, as are also the 
two planes passing through any normal and its two corresponding radii. 

Moreover a glance at fig. 2 will show that the two lines of vibration 
corresponding to any front lie respectively in the two meridian planes passing 
through the perpendicular to that front or, in other words, the intersection 
of a plane drawn through either ray belonging to a front perpendicular thereunto 
is always a line of vibration in that front. 

This has been noticed, I think, by Sir William Hamilton for the particular case 
of the singular points. 

As two fronts belong to every ray, so two rays pertain to every front. And 
from what has been said above it appears that the two lines of vibration in any 
front are the projections of its two rays upon its own plane. 



24 Analytical Development of [1 



Note 1. 

In the paper above, it is shown that the meridian plane, that is, 
the plane containing the ray and normal, always passes through a line 
of vibration in the corresponding point. Now the line of force called into 
action by a displacement in the line of vibration clearly lies in this very 
plane ; for the resolved part of it lies in the line of vibration itself. 

Harmony and analogy concur in suggesting that as two of these four 
lines are perpendicular to each other, so are also the other two, or in 
other words, that the ray is always perpendicular to the direction of 
unresolved force. 

The following investigation verifies this conjecture. 

Let X, y, z be the coordinates of a point taken at distance unity from the 
origin and in any line of vibration ; then the cosines of the angles made by 
the line of force with the axes are as a?x : b^y : c-z respectively. 

Let a be the inclination between the line of vibration and the line of 

force, then 

cv'x . X + Jfy .y + c'^z.z a-x^ + 6 V + o'z^ 

cos oy ^ ^ 

Vt(a*a;' + by + Cz-) {a? + 2/' + 2')) Vla'^' + iy + c"^) ' 

Let \/(aV + 6y + c^0-) = P, 

then P^ = v^ (sec wf. 

Now let a, ,8, 7 be the angles of inclination between the coordinate 
planes and the front in which the line of vibration lies, and X, some quantity 
to be determined. I have shown in Prop. 3 that if 

X cos a = {a? — v") x, 

then will A, cos /3 = (6^ — v^) y, 

and X cos <y = {c'^ — v^) z ; 

therefore \" = a^x- + 6y + cV — 2ti- {a-xr + If-y"- + c-z-") + t;** = P^ — v*. 

Again, 

X=.(,^. + ,^., + /|^.) = -= + ^^ + ^= = l; 
\(a^ — v'f {0^ - v^y (c^ — v-fj ^ 

therefore -J- = ^""^"^° + ^^^^ + ^""^y^^ 

Now -=- -, = -r^ r. -. = -; (cot CD )^ 

pa — y* ■y^ (see (of — V^ V^ ' 



1] FresneTs Optical Theory of Crystals. 25 

And in Mr Smith's investigation of the form of the wave surface (already 
alluded to*) by great good fortune I find ready to my hand 

(cos a.)- (cos ^y (cos 7)' _ 1 
{a" - v^y "^ (b^ - vy- {c- - v^ ~ v" (r^ - v") ' 

r being the radius vector to the point whose tangent plane is parallel to 
the point in question. 

Hence 

. ■y' v'' p'-' 

(cot coY = ^rn — n = -I — ; = -T^- ' 

p being the length of the perpendicular from the centre upon the tangent 
plane, for p = v. 

Hence (cot cof = the square of the cotangent of the angle between radius 
vector and normal. 

Or, in othe)' words, the line of force is as much inclined to the line of 
vibration as the ray is to the normal. 

Now the normal is perpendicular to the line of vibration, and all four 
lines lie in one plane. 

Therefore the ray is perpendicular to the line of force. Q. E. D. 

I may be allowed to conclude this long paper with a summary of some of 
the most remarkable consequences which I have extricated firom Fresnel's 
hypothesis. 

(1) The two meridian planes corresponding to any given radius are 
perpendicular to each other-f-. 

(2) So are the two corresponding to any given normal. 

(3) Every meridian plane bisects the angle formed by two planes drawn 
through the radius and the two prime radii. 

(4) It also bisects the angle formed by two planes drawn through the 
normal and the two prime normals. 

(5) Each meridian plane contains one line of vibration and the corre- 
sponding line of force. 

(6) The ray is perpendicular to the line of force. 

All these conclusions, except the fourth, are, I believe, original. 

* See above, p. 18. 

+ I have defined the meridian plane to be that which contains radius vector and normal 
belonging to the same point. 



26 Analytical Development of [1 

The theory of external and internal conical refraction follows immediately 
as a particular consequence from the third and fourth combined as already 
shown ; the same propositions also enable us to draw a tangent plane to any 
point of the wave surface by mere Euclidean geometry. May not some of 
these conclusions serve to suggest to physical inquirers the question, Has 
the theory been started from the most natural point of view? 



Note 2. Investigation* of the Wave Surface. 

Since the appearance of the preceding parts, I have succeeded in com- 
pleting the self-sufificiency of my method by deducing the equation to the 
wave surface from the expressions given in Prop. 5 for the angles between 
a front and the principal planes in terms of its two velocities. If these 
angles be w, (f), yjr, and the two velocities v^, Vn we found 

/{a^ — vi') (a- — vi) 
(a= - ¥) [a^ - c") ' 

/ (b'-v-')(¥-vJ') 



C0St = y/- 



(c- — Vj^) (c- — v^-) 



(c- — a") (c- — 6") 

Let the tangent plane to the wave surface be written 

COS o) cos A cos "v^ -, , s , 
.x-\ -.y-\ ^.^ = 1, (a)t 

, cos 6) I , cos <f) , cos ^p' 

then ~;-'x + ^2/+ r^^=0, (0) 

dC^X 4M c^f^^ 



d cos o) d cos d> d cos ylr ,. , . ' 

-, , x, « + J, Z y + -,, J z = Q. (7) 

^\/|^=^' Vl(c-a^)(c-6=)}4, 

* This investigation supplies the step which Mr Tovey was desirous should appear in the 
magazine. [Phil. Mag. March, 1838, p. 261. Ed.] 

t In lieu of I'j we might write v„ in the denominator without affecting the result. 

J. ^1, ii. ^cosw V \\v,- J^ -' , , ,, I 

± Observe, that = — — rr— r — . „ , „ s-: — , and so on for the rest. 



1] FresneVs Optical Theory of Crystals. 27 

then equation (7) becomes 

A^x + B7^y+Gi;z = 0, (1) 

and equation (;8) 

Aa? , Bh- , Cc- /9^ 

? V b 

and equation (a) may be written under two forms, viz. 

(a= - v,-) A^w + (¥ - n^) Bvy + (c= - v,') G^z=\, (3) 

S-')f"($-)f'-(S-)?-'' <*' 

From (1) 

A^x + Br^y = - C^z. (5) 

From (2) 

From (3) and (1) 

A{a:'-&)^x + B(b--c')riy=-l. (7) 

From (2) and (4) 

A{ar-c')% + B{h'-&)y-=<f. (8) 



From (5) and (6) 

G'-d'z" - R-lff' - A^a'x"- = ABxy (a?'^ + 6= ^ 
From (7) and (8) 
0= _ Br- {b"- - c^y f- - A' (a= - cj af = ABxy(l-h^)x (a' - c%b"- - c'). (10) 



(9) 



From (9) and (10) 
AB (a' - ¥) (a= - c=) (b"- - c") xy^= a?c- - (a- - c=) (6= - c-) C'c'^z' 

- {a' (b"- - cj - b' (a' - c=) (b' - c')} B'f 

- {a" {ct? - c-y - a" {a? - c") (b^- - c-)) A-x^ = a?c' - c^z'' - cy - aV. (11) 
From (11), interchanging {a, x, f) with (b, y, 1?) we have 

AB(b--ar) (b°- - (f) (a' - a") xy'^ = ¥c- - c'-z' - cV - by. (12) 
Finally, from (11) and (12) we have 
{a-c- - {a^ - C-) afl — c" (x- + 2/" + z^)} {b-c- - (b- — c=) y- - c" {x- -hy- -¥■ z')} 
= (a^ — c") {b- — c") afy-, 
that is (x^ + 2/" + z") (a"a? + by + c^z") — a- (b- + c") x" 

— b"- {a- + C-) y- — d' {b- + ar) z- + a-b'V = 
the equation required. 



ON THE MOTION AND REST OF FLUIDS. 
[Philosophical Magazine, xiii. (1838), pp. 449 — 453.] 

M. Ostrogeadsky's memoir oa this subject inserted in the Scientific 
Memoirs seems to have excited much attention, and has been made the 
occasion of some annotations* by a distinguished writer in the Philosophical 
Magazine. Mr Ivory's recent papers in the same periodical must still more 
tend to invest with a new interest all such speculations. It seems to me 
desirable therefore to present the theory of fluids in all the simplicity of 
which it is susceptible. 

I consider a fluid as a collection of particles subject to some law of 
relative position other than that of rigidity. These particles by their mutual 
actions maintain the connections of the system. As to the law of force 
between them we know nothing ; but I assume it is a general principle 
of nature, that for each instant of time the sura of the internal actions 
(reckoned by the product of each particle into the square of the space due 
to the internal force acting on it) is a minimum. This in fact is Gauss's 
principle of least restraint. We may if we please split this principle into 
two parts ; that is to say, assume that the internal system of forces is always 
such as if acting alone would keep the fluid at rest ; and then again assume 
that any equilibrating system of forces must be subject to the law of virtual 
velocities. I say assume, because it is impossible d priori to prove this. 

Lagrange's so-called demonstration is unworthy of his name, and (albeit 
sanctioned by the powerful oral authority of an ejc-Cambi'idge Professor) 
contrary alike to sense and honesty. It is better therefore at once to 
proceed upon Gauss's principle. It might easily be shown that this is in 
effect tantamount in all cases to D'Alembert's and Lagrange's principles 
combined. 

Before entering upon the investigation I may call attention to one point 
of great analytical interest, and relating to the difiicult subject of the 
algebraical sign, viz. that if the density of a point {x, y) in any circumscribed 

space be expressed by the quantity t- + t- so that the mass is 



//'^^'^KS+iJ^^^KI)' 



[* PUl. Mag. May, 1838, p. 385. Ed.] 



2] On the Motion and Rest of Fluids. 29 

that is not equivalent to 



that is if we please 



{udy + vdx), 

dy dx\ , 
ds dsj 



(where s is for clearness' sake and to avoid double limits taken an element of 
the bounding curve) as at first sight it might appear to be, but is in fact 

equal to 

dy dx\ -, 
u ^—v-j-) ds. 
as ds) 

I shall demonstrate this point in the next number* of the Magazine. 

It at first caused me some trouble in conducting the annexed inquiry. 

I shall also take occasion at some other time to revert to a new species 

(as I believe) of partial differential equations ; that is to say, where there 

are fewer of them than of the principal variables, which may be called 

therefore Indeterminate Partial Differential Equations. A complete solution 

of one of these appears in the subjoined 

Investigation. 

For the sake of simplicity I take an incompressible fluid. The method 
is nowise different for a fluid of varying density. 

Let Aa;, Ay, A? be any displacement undergone by a particle at the 
point X, y, z parallel to the axes x, y, z respectively ; it is easily shown that 
to satisfy the condition of invariability of mass we must have 
cZAa; dAy d^z __ 

dx dy dz ' ^ ' 

One relation between u, v, w the velocities parallel to x, y, z is obtained 
immediately by putting uht, vht, wBt, for A,x, Ay, Az, which gives 

du dv dw _ . 

dx dy dz ' ^ ' 

as usual. 

Again, if X, Y, Z be the impressed forces, and Xi, Fj, Z^ the internal 
forces acting on any particle parallel to the axes, we have 

„ _ dw du du du 

^^ + ^=dt+Tx'' + Ty' + d^'"' ^2) 

„ „ dv dv dv dv 

^^^^-dt^Tx'^^dy'^dz'"' (^) 

„ „ dw dw dw dw ,,. 

^^ + ^ = W^^"+^^ + ^"'' (*) 

from the mere geometry of the question. 

[* p. 36, below. Ed.] 



30 On the Motion and Rest of Fluids. [2 

Finally, Gauss's principle teaches us that 

dxdydz [X,^.^,+ \\^Y^ + Z,IS.Z;\ = 0. (/3) 



^ rf(Z + ZO , d{Y+Y, ) , d{Z + Z,) 

JN ow J 1 J H } 

dx dy dz 

Idu^ fdvV' /dw\~ \dv dw dw du du dv 
"^ \dxj \dyj \dzj \dz dy dx dz dy dx 

as appears from the equations (1), (2), (3), (4) ; and hence 

d^X^ dAY, d ^Z, 
dx dy dz ' 

the complete solution of which, free from the sign of integration, is 

AZ = '^ - ^ 
^ dy dz' 

^ dz dx' 

A^ = '^ - — 
' dx dy ' 

CO, (j), i/r being any three independent functions of x, y, z. 

On substituting these values in equation C/3) we obtain 

This may be put under the form 

dzjjdxdy[^^i^X.)-lifY.)] 

+ jdxjjdydz\^^^{o>Y,)-^(coZO 



+ jdyijdzdx\^^i4>Z,)-^^{ct>X,) 
-jjjdxdydz.col^^^^-"^^^'^ 



-jjjdxdydz . 4> C^ - '^] = 0. 



dy, 

iX, 
dx dz J 



2] 



On the Motion and Rest of Fhdds. 



31 



Here it must be remembered that co, <^, ->|r are perfectly independent of each 
other. Also the values of the three first written quantities depend upon the 
values of X^, Fj, Z^ at the bounding surface; the values of the three last- 
written depend upon the general values of X^, Y^, Z^. It is clear therefore 
that each system of three equations and each member of each system must 
be separately zero. 

The three latter equations give 

dX, _ dY, ^ ^\ 
dy dx 

dY_dZ,_ y 

dz d' ^ 



dZ, _ dXy 
dx dz 



= 



(7) 



The three former require that for each section of the surface parallel to 
the plane xy 

l^{X4x+Y^dy)=Q, 

for each section parallel to yz 

1 0) (Y^dy + Z,dz) = 0, y , 



for each section parallel to zx 



I <l)(Z,dz + X4a:)=0 . 



(8)* 



and these equations are to hold good whatever if-, (p, oo may be. From the 
equations (7) we derive 

X,dx + Y,dy + Z,dz = df, (5) 

from equations (8) we obtain 

f= constant for all points in any section of the bounding surface parallel 
to the plane of xy, 

/= constant for all points in any section of the bounding surface parallel 
to the plane of yz, 

y = constant for all points in any section of the bounding surface parallel 
to the plane of zx. 

Now by drawing through all the points in a plane parallel to xy, planes 
parallel to yz, we may cover the whole surface ; hence / is constant all over 
the surface bounding the fluid. 



See remark at introduction. 



32 071 the Motion and Rest of Fluids. [2 

Therefore X^ dx+Y-,dy + Z^dz = 0, (6) 

for all variations of dx, dy, dz taken upon the surface. 

The equations (1, 2, 3, 4, 5, 6) are coincident with those obtained by the 
usual method; with this difference, that X^, Y^, Z^ here take the place of 

dp dp dp 
dx' dy' dz' 

Thus then we have obtained all the conditions requisite for determining 
the motion of fluids from the universal principle of least constraint conjoined 
with the specific character of the system in question. 



General Remarks. 

In the case of equilibrium, that is in the case where no particle moves, 
we have X, + X = 0, Y, + T = 0, Z, + Z=0. Hence Xdx+Ydy+Zdz is a 
complete differential always and zero for the surface. 

The above results have been obtained upon the principles of the differ- 
ential calculus, and the continuity of the forces has been tacitly assumed. 
If now we were to suppose forces of finite magnitude (as compared with the 
whole sum acting tipon the entire s}'stem) to be applied to a layer of single 
particles or to a layer of a thickness of the same order of magnitude as the 
distances between the particles themselves, (which has been treated as an 
infinitesimal) it would appear that our results would be no longer applicable, 
just in the same manner as it would be erroneous to apply the principle 
oi vis -viva (for example) without modification, to the case of impulsive forces, 
because we had deduced it by the calculus in the case of the motion 
being continuous. Hence the above equations ought not strictly to apply 
to the motion or rest of a fluid contained between physical surfaces ; for the 
pressure afforded by these surfaces, whatever its actual value may be, we 
know d priori is commensurable with the whole amount of force acting on 
the fluid ; but the immediate application of this pressure (alias repulsive 
force) is confined to the bounding layer of fluid particles, or at most extends 
to a distance bearing a low ratio to the distances between the particles 
themselves. 

Accordingly, to the non-applicability of the equations for free fluids to the 
case of fluids confined at the boundaries, and to an independent investigation 
upon the minimum principle for this class of problems, it is, that I look for 
the true explanation of the phenomena of capillary attraction (vulgarly so 
called). 



ON THE MOTION AND REST OF RIGID BODIES. 

{Philosophical Magazine, xiv. (1839), pp. 188 — 190.] 

In the subjoined investigation, which, as far as I know, is my own, 
I apply the same method to rigid as in the preceding paper I applied to 
fluid systems. 

Let X, y, z be the coordinates of any particle in a rigid body; x' , y , z! the 
coordinates of some other particle, and let 

x' =x + h, y' = y + k, z' = z + I. 

Call Ax, Ay, Az the increments which x, y, z receive after the lapse of a 
small interval of time ; so that terms in which they enter in two or more 
dimensions may be neglected. 

mi * , /^ A ^^x 1 d^^ 1 dAx , , „ 

Then A («') = Ax + -^— h + -j— k+ ^r-^ + P> 

^ ax ay az 

. , ,, . dAz , dAz , dAz , „ 
A (z') = Az + -T- h + —j— k + —J- ^ + ^' 
- dx dy dz 

P, Q, R containing binary and higher combinations of h, k, I, which we shall 
have no occasion to express. 

At the commencement of the interval the squared distance of the two 
particles was (x' — x)- + {y — yf + {z' — zY ; at the end of the interval the 
distance squared is 

(x'-x + A (x) - AxY + {y' -y + A (y) - Ayf + {z' -z + A {z') - Azf, 

and these two expressions must be the same by the conditions of rigidity 
whatever h, k, and I may be ; that is 

, „ ,„ / , dAx , dAx , dAx , ^X- 
h^ + J^ + l^={^h+-^h+^k + -^l+P) 

+ (k + ^h + '^4yk + '^-^l + Q 
\ dx dy dz 



I , dAz , dAz , dAz , , „V 



dy 
for all values of h, k, and I. 



dx 


(a) 


dy 


(&) 


^f =0- 
dz 


(c) 



34 On the Motion and Rest of Rigid Bodies. [3 

Hence rejecting infinitesimals of the second order and equating to zero 
separately the coefficients of A^, k'^, 1-, and of kl, Ih, hk, we have 

dAw dl^z ,^ , ,, 

dl^z d!^x _ 
dx dz ' ' 

By differentiating (cZ), (e), (/') with respect to z, x, y respectively, and 
substituting from (a), (&), (c), we obtain 

cZ^Ay _ c?-Ag _ (i^^a; _ 
cZ^;- ' dx" ' dy'^ 

By differentiating the same with respect to y, z, x respectively, and pro- 
ceeding as before, we have 

d=A2 _ d-^x _ d^'Ay _ 
dy- ' dz" ' dx^ 
Thus, then, we have 

dAx _ d^Ax _ d^Ax 
dx ' dy" ' dz^ ' 
dAy^Q drAy^^ d'Ay^^ 
dy ' dz" ' dx^ ' 
dAz _ d^Az _ d^Az _ 
dz ' dx'' ' dy^ ' 
therefore Ax = A + By + Cz, (o) 

Ay = D + Ez + Fx, (p) 

Az = G + Hx + Ky, (?) 

A, B, G, D, E, F, being constant for a giveii instant of time; between which 
by virtue of the equations (d), (e), {/), we have the relations 
E + K = 0, H+C = 0, B + F = 0. 
If we call u, V, w the three component velocities of the particles at x, y, z 
parallel to the three axes, and Xj, Y-^, Z^, the three internal forces, it is at 
once seen that xi, v, w, as also AX^, AYj, AZ^ must be subject to the same 
equations as limit Ax, Ay, Az ; so that 

u = a + yy — ^z, (1) 

v = b + az — yx, (2) 

w = c + ^x — ay, (3) 

' AX, = a, + y,y-^,z, (h) 

AY, = b,+a,z-y,x, (j) 

AZ^ = Ci + A« - "ly- W 



3] 



On the Motion and Rest of Rigid Bodies. 



35 



Also if X, Y, Z be the impressed forces, we have 

du 



X, + Z = 



dt' 



Y + Y-- 
dw 



Z, + Z = 



dt 



(4) 
(5) 
(6) 



And by Gauss's principle, calling m the mass of the particle at x, y, z. 

Hence equating separately to zero the coefficients of ttj, \, Ci and of 
«!, A> 7i i^i the quantity 2m(XiAZi+ FiAFj + Z-^I\Z^ we have 

2??i Xi = 0\ 
2mFi=0 
Sm ^1 = 

2m {X^z - ^la;) = 
%m{Y^x-X^y) = 0, 
Lastly, we have the equations 



)■■ 



(7-12) 



dx 

- = t' 
dt 

dz 

'' = df 



(13) 
(14) 
(15) 



From the fifteen equations marked (1) to (15), the motion may be deter- 
mined by assigning the position of each particle at the end of the time t in 
terms of its three initial coordinates, its three initial velocities, and the 
initial values of the nine quantities 

1,mx, l,myz, Xmx\ 

Xmy, "Zmzx, "^iny^, 

%mz, "^mxy, 'Zmz-. 

In the case of rest Xi = — X, Y^ = —Y, Z^ = —Z, and the equations (7) 
to (12) inclusively taken, express the conditions of equilibrium. 

The equations (o), (jo), (g), which have been obtained from conditions 
•purely geometrical, establish the well-known but interesting and not obvious 
fact, that any small motion of a rigid body may be conceived as made up 
of a motion of translation and a motion about one axis. 

3—2 



4 



ON DEFINITE DOUBLE INTEGRATION, SUPPLEMENTARY 
TO A FORMER PAPER ON THE MOTION AND REST 
OF FLUIDS. 



[Philosophical Magazine, xiv. (1839), pp. 298—300.] 

In a paper on Fluids which appeared in the December Number of this 
Magazine, I had occasion to remark, that the mass of an area having at the 

point (oo, y) a density t- + t- could be expressed by the simple formula 



/:{• 



as as) 



I being the length, and ds an element of the bounding curve : this may be 
thought to require some explanation. 

(1) 



Let APBq represent any oval; PpL, QqM any two contiguous 
ordinates cutting the curve in Pp, Qq 
respectively, A G, BD the two extreme 
tangents parallel to Oy, and p the 
density at any point {x, y). The ex- 
pression JJpdxdy will serve to denote 
the mass of the oval area APBq, and 
the limits may be twice taken, that is, 
(i) the two values of y corresponding 
to any one of cc; and (ii) the two 
values of x corresponding to C and D. 
This method is in fact tantamount to 
taking the sum of the columns Pp 
qQ ; but this is not necessary, for 
APBq may be considered as the algebraical sum of the mixtilinear area 
APQBDC, and the mixtilinear area BDCApq, or (if any line O'G'D' be 
drawn parallel to OCLMD) of APQBD'C and BD'C'Apq. 

Thus then the mass = jda; (jpdy), Jpdy being left indeterminate, and the 
extremity of x travelled round from G to D, and back again from D to C. 




Fig. 1. 



4] On Definite Double Integration. 37 

This will be better expressed by transforming the variable, and summing 
with respect to some quantity, such as the arc of the curve, which contin- 
uously increases, or if we please, with respect to 6, the angle subtending any 
point taken within the curve. 



The mass is then 



= ±£c^. {(/,.,)§; 



always remembering that no constant need be added to jpCiy, and that the 
doubtful sign arises from the chbice of ways in which 6 may be measured 
round. If the area be not included by one line; but by several, as for 
example, by a curve and a right line, the above integral, if broken up into 
as many parts as there are breaches of continuity, will still apply. 

(2) Let us suppose that we have two areas exactly coinciding with, and 
overlapping one another ; but the density of the one at {x, y) to be p, and of 
the other p . 

Let the mass of the first be treated as the sum of columns parallel to Oy, 
and that of the second as the sum of columns parallel to Ox. 

The one will be represented by 

±\jei^P^y)%, 
the other will be represented by 

± f dO (Jp'dx) ' 
and the sum of the two, or the joint mass, by 

So long as these two operations are performed separately, the doubtful 
signs may be preserved in each term, because s need not be travelled round 
in the same direction for the two summations ; but if we perform the second 
integration conjointly for the two masses, their sum 

= ±Jlde\(!pdy)f^±HIp'dx)% 

the mark of interrogation denoting that one or the other, but not either of the 
signs + must be used, and the question is, which ? 

This will be answered by taking different points in the bounding line 
which may be continuous or not. Now every line returning into itself, 
whether continuous or not, will naturally divide with respect of any given 



dy 
W 



38 



On Definite Double Integration. 



[4 



system of axes, into at most four parts, or sets of parts ; two in which dx and 
dy both increase or both decrease, and two in which one increases and the 
other decreases. 

Take Pj, P^, P^, P4, any points in the four quadrants respectively, it 

will be observed that. 

At Pi the p column enters ad- 
ditively, and the p column subtrac- 
tively. 

At P„ both columns are additive. 
At P3 the p column is additive 
and the p column subtractive. 

At P4 both columns enter subtrac- 
tively. 

Again, reckoning round in the 
direction of the arrows. 
At Pi, a; and y are both increasing. 
At P2, a; is increasing and y decreasing. 
At P3, a; and y both decrease. 
At P4, a; is decreasing and y increasing. 

Thus when ^pdy and ^p'dx are affected with the same signs, dx and 
dy are of opposite signs ; and when jpdy, Jp'dx are of opposite signs, dx and 
dy are of the same sign. 

Hence it appears that the mass of the area, whose density at {x, y) is 
p + p', is capable of being represented by 




Fig. 2. 



4 

J '2,11 



de 



Updy)%-Up'dx)f^ 



b. 

ON AN EXTENSION OF SIR JOHN WILSON'S THEOREM 
TO ALL NUMBERS WHATEVER. 

[Philosophical Magazine, XIII. (1838), p. 454.] 

The annexed original theorem in numbers will serve as a pendant to 
the elegant discovery announced by the ever-to-be-lamented and com- 
memorated Horner*, with his dying voice, in your valued pagesf. 

Theorem. 
If N be any number whatever and 

PuP«.,Ps Pc 

be all the numbers less than N and prime to it, then either 

Pl-P2-Pi Pc+1, 

or else Pi-P^^p^ Pc — ^, 

is a multiple of N. 



NOTE TO THE FOREGOING. 

[Philosophical Magazine, xiv. (1839), pp. 47, 48.] 

I HAVE to apologize for calling " original " (in the last Number of the 
Magazine) the theorem of numbers which I termed " a pendant to Horner's 
theorem." This Mr Ivory has done me the honour to inform me may be 
found in Gauss's Disquisitiones Arithmeticce, p. 76. As Horner's extension 
of Fermat's theorem suggested this extension of Sir John Wilson's to me, 
so I concluded that had this extension of Wilson's been known to the world 
it would naturally have suggested his to Horner. No acknowledgment of 
this kind having been made, I took it for granted that the theorem I gave 
was new. Undoubtedly had Mr Horner been aware of Gauss's theorem 
he would have made mention of it. 

I take this opportunity of adding that my acquaintance with Gauss's 
principle:): has not been derived from the study of his works, but from a 
casual statement of it in an English work. Dynamics, by Mr Earnshaw, of 
St John's College, Cambridge. 

* Homer's proof is highly valuable as a novel and highly ingenious form of reasoning, but 
his theorem may be deduced with infinitely more ease and brevity from Fermat's than he seems 
to have beerl aware of. 

[t Phil. Mag. Vol. xi. p. 456. Ed.] [J See p. 28 above. Ed.] 



7. 



ON RATIONAL DERIVATION FROM EQUATIONS OF COEXIST- 
ENCE, THAT IS TO SAY, A NEW AND EXTENDED THEORY 
OF ELIMINATION*. PART I. 

[Philosophical Magazine, xv. (1839), pp. 428 — 435.] 

Ant number of equations existing at the same time and having the 
same quantities repeated, may be termed equations of coexistence : in the 
present paper we consider only the case of two algebraical equations : 

ic™ + Oia;"^! + a^x"'-- + + a™ = 0, 

«"+&!««-' + hx'"--- + + bn = 0. 

The above being "equations of coexistence," x is called "the repeating term." 

If we suppose the equation 

Co*'' + Cja;''"^ + c,*''"'^ + + c,. = 

to be capable of being deduced from the two above, and, therefore, necessarily 
implied by them, this will be called "a Particular Derivative" from the 
equations of coexistence, of the rth degree, (r being supposed less than m 
and nf , and the coefScients being rational functions of the coefficients of the 
equations of coexistence). 

There will be an indefinite number in general of such derivatives, and 
the form involving arbitrary quantities which includes them all is called 
" the general derivative of the rth degree." 

Any "Particular Derivative," in which the terms are all integral, 
numerically as well as literally speaking, is called an " Integral Derivative." 

That " Integral Derivative " of any given degree in which the literal 
parts of the coefficients are of the lowest possible dimensions*, and the 
numerical parts as low as they can be made, is called the " Prime Derivative " 

[* The results of this and some following papers were repeated, with demonstrations, in the 
paper " On a Theory of the Syzygetie Relations of two rational integral functions comprising an 
application to the Theory of Sturm's Functions, and that of the greatest Algebraical Common 
Measure," Phil. Trans. Royal Soc. Vol. csLni., Part i. pp. 407 — 548, 1853. See below Section n. 
Art. (16) of that paper. Ed.] 

t This restriction upon the value of r is not essentially requisite, and is only introduced 
to keep the attention fixed upon the particular objects of this first Part. 

t Of course the dimensions of the coeflScients in the equations of coexistence are to be 
understood as denoted by the indices subscribed. 



7] On Rational Derivation from Equations of Coexistence. 41 

of that degree. So that there is nothing left ambiguous in the prime 
derivative save the sign. 

The " Derivative by succession " is that particular derivative which is 
obtained by performing upon the equations of coexistence the process 
commonly employed for the discovery of the greatest common measure, and 
equating the successive remainders to zero. 

To express the product of the sums formed by adding each of one row of 
quantities to each of another row, we simply write the one row above the 
other ; a notation clearly capable of extension to any number of rows, which 
would not be the case if we spoke oi differences instead of sums*. 

Theorem 1. 

Let hi, h„, ... hm, be the roots of one equation of coexistence, ki, k^, ... kn, 
the roots of the other. The general derivative of the rth degree is repre- 
sented by 

2 (^SR (h, , h, h... K) {{x - K) (x-h,)...{x- K)} X 1^+^' ^+^^ '"^Zk])^^' 

SR (Ai, /i2, As ... K) denoting any symmetrical rational (integral or fractional) 
function of Aj, Aj . . . A^ ; 

J Ar+i ) A^_|_2 • • • Am 1 
(— Aj , — k2 ... — fCnJ 

being to be interpreted as above explained, and 2 of course including as 
many terms as there are ways of putting n things r and r togetherf . 

A form tantamount to the above, and which may be substituted for it, 
is its analogue, 

i(^sR(k,,h...kr){ix-h)(w-k;)...(x-kr)}x^^_;;y^^''_;;-*-^^ 

When r = the theorem gives simply 

(A.A....A™ ]^^ 

{ — n-i ) — K^ ... — Kn) 

and is coincident with that given by Bezout in his Theory of Elimination. 

* The wider views which I have attained since writing the above, and which will be developed 
in a future paper, lead me to request that this notation may be considered only as temporary. 
It would have been more in accordance with these views to have used the two rows to denote 
products of differences than of sums. But a change now in the text would be very apt to cause 
errors in printing. 

t The general derivative may clearly be expressed also by the sum of any two particular 
derivatives affected respectively with arbitrary rational coefficients. The equivalency of an 
arbitrary function to two arbitrary multipliers is very remarkable, and analogous to what occurs 
in the solution of certain differential equations. 



42 On Rational Derivation [7 



Subsidiary Theorem (A). 

If /ti, ^2 ... hm be the roots of the equation 

X™ + Oi*'""' + ttoa;™"^ + + a,„ = 0, 

and if e™ + aje'"-^ + 0,6"'"^ + + aOT — m = 0, 

*^en :S rr , , ,, Y, T^ ^, = -^ /- 2 (e'-+0, 

(fti - /io) (fti - A3) ... (/ii - /im) r + 1 c^M ^ ^ 

M being made zero after differentiation. 

CoK. If iJ(/ii) denote any integral rational function of Ai, then 

2 Mhl 

(Ai - Z^) (Ai - A^) ... Qh-h^n) 

is always integral and is zero when the dimensions of RQh) fall short of 
(m-1). 

Subsidiary Theorem (B). 

^ SR{K,h,...hr) 
j Aj , h^ ... hr 

(— «r+i , — "'^+2 • • • A„ 

can be expressed by the sum of terms, each of which is the product of series 
of the form 

^ RiKl 

"' (Ai - Aj) (h^~h,)...{h^-hn)' 

it is always integral, and when the dimensions of the numerator fall short of 
(m —r)r it vanishes*. 

Subsidiary Theorem (C). 

The only modes of satisfying the equation 

2 {f{h„ h^...hr)x SR Qh,K... K)] = 0, 

for all forms of the latter factors short of (m — r)(n — r) dimensions, are to 

put/(Ai, A2 ... /i,.) = 0, or else 

.,, , , , constant 

/(Ai, h.,...hr) = 



Ai, h«...hr 



* It may be remarked also in passing, that any term in the numerator which contains any 
one power not greater than m- 2r may be neglected and thrown out of calculation. Moreover, 
an analogous proposition may be stated of fractions in the denominators of which any number 
of rows are written one under the other ; see the first note, page 41. 



7] from Equations of Coexistence. . 43 

Theorem 2. 
By virtue of the subsidiary theorem (B), the two equations 



"-r+i) "r+2 ••• "•m I / 
-h„ -K.-.-hrSj 

"'r+l > n-r+a ■•• Kn I \ 



\ J "-r+l ) "■r+2 ■■• Kn I / 

\ 1-^1, -fc...-W 

are each integer derivatives of the rth degree. 

Theorem 3. 

And by virtue of the subsidiary theorem (C), the two above equations 
are the " Prime Integer Derivatives," and are exactly identical with each 
other. 

Cor. 1. The leading coefficient of the "prime derivative" of the rth 
degree is always of (m — r) (n — r) dimensions. 

COE. 2. If P,. be the prime derivative of the rth degree and if 
(X=0, F=0) be the two equations of coexistence, and X^, fi^r the two "prime 
constituents of multiplication" to the said derivative, that is if X^ and /u.^ 
satisfy the equation Xr^ + f^ry= -Pr, then the coefficient of the leading 
terms in V and in /x^ is of (m — r — l)(n — r—l) dimensions. 

Theorem 4. 

The " Prime Derivative " of any given degree is an exact factor of the 
" derivative by succession," of the same degree. The quotient resulting from 
striking out this factor is called " the quotient of succession." 

Theorem 5. 

If Zi, Z2, X3, &c., be the leading coefficients of the derivatives occurring 
first, second, third, &c., in order after the equations of coexistence, and if 
Qi> Q2, Qs, fee., represent the first, second, third, "quotients of succession" 
reckoned in the same order, then 

Q. = 1, 
-^'' 



44 On Rational Derivation [7 

and in general 



Z(2 jL'4 . . . -t/2)J— 4 -^21 

2 
11— 1 



' T,*T,* T, * 

T 41 i T *T ■» 

Mji u^ . . . i-ian—s ^^n—i 



X/o X/4 . . . jL/27l— 2 -^25 

Cor. Hence, in place of Sturm's auxiliary functions, we may substitute 

the functions derived from the equations of coexistence [fx = 0, -^ = 

according to Theorem 2, due regard being had to the sign. 

Scholium. Hitherto it has been supposed that the values of the coeffi- 
cients in the equations of coexistence are independent of one another, but 
particular relations may be supposed to exist which shall cause the leading 
terms given by Theorem 2 to vanish, giving rise to anormal or singular 
primes, as they may be called, of the degree r of fewer than (m — r){n — r) 
dimensions. The theory of this, the failing case (so to say), is highly interest- 
ing, and I have already discovered the law of formation for the quotients of 
succession on the supposition of any number of primes vanishing consecutively; 
but I forbear to vex the patience of my reader further, the more so, as I 
hope soon to be able to present a complete memoir, with all the steps here 
indicated filled up, and numerous important additions, (the perfect image 
of which this is but a rough mould), as homage J;o the learned and illustrious 
society which has lately done me the honour of admitting me into its ranks. 

Why this has not already been done must be excused, by the fact of the 
theory having suggested itself abroad in the intervals of sickness|. Yet thus 
much will I add in general terms, namely, that as many primes as vanish 
consecutively, so many units must be added to the index 2 of the accessions 

* That the appearance of the index 4 may not startle, let my reader bear in mind that there 
are what may be termed secondary derivatives of succession for every degree appearing in the 
process of successive division. 

t The prime derivatives must be capable of yielding an internal evidence of the truth of 
Sturm's theorem. In fact, for the case of all the roots being possible, a little consideration will 

serve to show that the leading term of each prime derivative of the equation \ fx-j—l =0 will 

consist of a series of fractions, each of which fractions is, mimerically speaking, of the smne sign. 
+ The reflections which Sturm's memorable theorem had originally excited, were revived by 
happening to be present at a sitting of the French Institute, where a letter was read from the' 
Minister of Public Instruction, requesting an opinion upon the expediency of forming tables of 
elimination between two equations as high as the 5th or 6th degree containing one repeating 
term. The offer was rejected, on the ground of the excessive labour that would be required. 
I think that this has been very much overrated ; and probably many will be of the same opinion 
who have dwelt upon the fact that no numerical quantity will occur in the result higher than 
the highest index of the repeating term. Would it not redound to the honour of British science 
that some painstaking ingenious person should gird himself to the task ? and would not this be 
a proper object to meet with encouragement from the Scientific Association of Great Britain ? 



7] from Equations of Coexistence. 45 

received in the numerator and denominator of the subsequent quotient ; 
and in the quotient after that, it is not the square of the leading term of 
the penultimate prime, — but the product of this term by the leading term 
of that anormal prime of the same degree which has the lowest dimen- 
sions, — that finds its way into the numerator. The rest of the formation 
remaining undisturbed, unless and until a new failure have taken place. 

Note on Sturm's Theorem. 

When one of the equations of coexistence is the differential coefficient 
with respect to the repeated term of the other, the prime derivatives given 
in Theorem 2 which coincide in this case with Sturm's auxiliary functions 
reduced to their lowest terms, may be exhibited under an integral aspect. 

Let SPD intimate that the squared product of the differences is to be 
taken of the quantities which follow it. 

Let Si indicate the sum of the quantities to which it is prefixed. 

82 the sum of the binary products. 

S-i the sum of the ternary products, and so on ' 
Let hj, h„...hn be the roots of any equation. 
Then Sturm's last auxiliary function may be replaced by 

8PD{hi,h„...K). 
The last but one may be replaced by 

tSPD {h„ }u_... /i„_0 X + tSi {h,, h,... K-i) SPD (hi,h,... K-i). 
The one preceding by 
ISPD ih,ho_... A^O «:' + S/Si (Ai ,h,... K-^) SPD {K,h... A^^) « 

+ ts^{h„h...K^) SPD {Jh,h,... K-.X 

and so on. 

Thus then Sturm's rule for determining the absolute number of real 
roots in an equation is based wholly and solely upon the following 

Algebraical Proposition. 

If there be n quantities, real and imaginary, tl imaginary ones entering 
in pairs, as many changes of sign as there are in the terms 
1, SPD {hi, h^), 
'ZSPD{hi,h„h), 



l.SPD{hi,\...hn-i), 
'ZSPD(hi,h,...hn), 

so many in number are these pairs. 



46 On Rational Derivation from Equations of Coexistence. [7 

Query (1). Is there no proposition applicable to any n quantities 
whatever ? 

Query (2). Is there no faintly analogous proposition applicable to higher 
powers than the squares ? 

Query (3). Seeing that in forming the coefficients in the equation of 
the squares of the differences, we pass from n functions of the roots to 

n — ^ — and not n functions, of their squared differences, does not a natural 

passage to the former lie through n functions of the squared differences ? 

In other words, may not the quantities tSPD Qi-i , k^... hn), &c., serve as 
natural and valuable intermediaries between the coefficients of an equation 
involving simple quantities and the coefficients of the equation involving the 
squares of their differences ? 

P.S. In the next part I trust to be able to present the readers of this 
Magazine with a direct and symmetrical method of eliminating any number 
of unknown quantities between any number of equations of any degree, by 
a newly invented process of symbolical multiplication, and the use of co7n- 
pound symbols of notation. 

I must not omit to state that the constituents of multiplication X^ and 
^,. explained in Cor. 2 to Theorem 3 are equal to the expression 

/^i, A'o ... kn-r-i 

S (a; - \) {x - L) ...{x- K-r-i) ^'' '^^ _ _' k'n-r-, 

V "-n— r • • • "'n 

and its analogue respectively. 



ON DERIVATION OF COEXISTENCE. PART II. BEING THE 
THEORY OF SIMULTANEOUS SIMPLE HOMOGENEOUS 
EQUATIONS. 

[Philosophical Magazine, xvi. (1840), pp. 37 — 43.] 

Art. (1). We shall have constant occasion in this paper to denote 
different quantities by the same letter affected with different subscribed 
numerical indices. 

Such a letter is to be termed a " Base." 

Every character consisting of a base and an inferior index, this index 
is called an argument of the base, namely, the first, second, or nth 
argument, according as 1, 2, or in general n, be the number subscribed. 

Art. (2). I use the symbol PB to denote the product of the differences 
of the quantities to which it is prefixed (each being to be subtracted from 
each that follows); thus 

PB (a, h, c) indicates (b — a) (c — a) (c — b). 

PB (0, a, b, c) indicates abc (b — a) (c — a)(c — b). 

PB (0, a,b,c...l) indicates abc ... I x PB (a,b,c... I). 

Art. (3). For want of a better symbol I use the Greek letter ^ to denote 
that the product of factors to which it is prefixed is to be effected after a 
certain symbolical manner. This I shall distinguish as the zeta-ic product. 

The symbol f will never be prefixed except to factors, each of which is 
made up of one or more terms, consisting solely of linear arguments of 
different bases, that is, characters bearing indices below but none above. 

I am thereby enabled to give this short rule for zeta-ic multiplication: 
" Imagine all the inferior indices to become superior, so that each argument 
is transformed into a power of its base ; multiply according to the rules of 
ordinary algebra ; after the multiplication has been done fully out depress 
all the indices into their original position ; the result is the zeta-ic product*." 

* It is scarcely necessary to add that an analogous interpretation may be extended to any 
zeta-ic function whatever. Thus 

f (ai + 6i)2=a2 + 2ai6j + fcj, 

fcosK) = l-j^+j-254_,<fee. 



48 On Derivatio7i of Coexistence. [8 

Thus for example ^(a^, bs) is the same as simply Urbs, but f (a^, ag) 
represents not ards but a^+s. 

So in like manner 

?{{an-b,)(ai-bj} 

= aji+i - dhbrn -bkai+ b^+k, 

? ((Oi - 61) («! - Ci) (&i - Ci)} 

= the depressed product of (a — 6) (a - c) (6 — c) 
= the depressed value of a- (6 — c) + 6^ (c — a) + c^ (a — 6), 
that is, = aa^i — a^Ci + 62C1 — 62"^ + CaOi — C261 . 

Art. (4). We shall have occasion in this part to combine the two symbols 
.^, PD : thus we shall use 

^ PD (ai^i) to denote ^ (61 — a-^, 

^PB (Oi^iCi) to denote ^ {(61 — a^ (cj — aj) (ci — 61)}. 

Art. (5). For the sake of elegance of diction I shall in future sometimes 
•omit to insert the inferior index when it is unity; but the reader must 
-always bear in mind that it is to be understood though not expressed. 

I shall thus be able to speak of the zeta-ic product of such and such bases 
mentioned by name. 

Art. (6). We are not yet come to the limit of the powers of our notation. 
The zeta-ic product of the sum of arguments will consist of the sum of 
products of arguments, «ach argument being (as 1 have defined) made up of 
a base and an inferior index. Now we may imagine each index of every term 
■of the zeta-ic product after it is fully expanded to be increased or diminished 
by unity, or each at the same time to be increased or diminished by 2, or each 
in general to be increased or diminished by r. I shall denote this alteration 
by affixing an r with the positive or negative sign to the ^. Thus 

f (oi — &i) («! — Ci) being equal to a^ — ajCi -h &iCi — 61 ai, 

^+1 («! — bi) («! — Ci) is equal to a^ — a^c^^- b^c^ — b«a^, 

f_i (tti — 61) (cfi — Ci) is equal to a, — aoCo -I- 60 Co — &o«o- 

In like manner i^PB (a, b, c) indicating 

b^i — bnCi + C2&1 — Cstti -I- a^Ci — aj)!, 

X±rPB (a, 6, c) indicates 

02±j.(Xi±)- t)2±r^i±r "1 C2±7-0i±r d^yOirkr "T" Ct2±r^i±r Oj<^dcr^\±r * 

I shall in general denote ^+rPB {a, b, c ... I) actually expanded as the 
:zeta-ic product of «, b^ c, ...I in its rth phase. 



8] On Derivation of Coexistence. 49 

Art. (7). General Properties of Zeta-ic Products of Differences. 

If there be made one interchange in the order of the bases to which 
^ is prefixed, the zeta-ic product, in whatever phase it be taken, remains 
unaltered in magnitude, but changes its sign. 

If in any phase of a zeta-ic product two of the bases be made to coincide, 
the expansion vanishes. 

Let Ji be used, agreeably to the ordinary notation, to denote the sum of 
the quantities to which it is prefixed, J2 to denote the sum of the binary 
products, /a of the ternary ones, and so on. 

Thus let /i (a-J)iCi) or Ji (a, b, c) indicate aj -)- 61 -t- Ci, 

and /a (ai^iCi) or /j {a, h, c) indicate ajfti -I- Orfii + h-^Ci, 

and /a (ai^iCi) or /s (a, h, c) indicate cbyb^d, 

we shall be able now to state the following remarkable proposition connecting 
the several phases of certain the same zeta-ic products. 

Art. (8). Let a,h, c, ... I, denote any number of independent bases, say 
(11— 1); but let the arguments of each base be periodic, and the number of 
terms in each period the same for every base, namely n, so that 

Qjy ^ d^J^yi = Ojf—^i , diji ^ OjQ ^ d — 7i , 

h,. = hr^n = K-n, bn = h= b-.n, 
Cy = Cj-4-)i = Cj._ji, C71 = Co ^ 0—tij 



r being any number whatever. Then 

^-,PD (0, a, 6, c ... = ? {/i {a, h,c... I) tPD (0, a,b,c... I)}, 
^_,PD (0, a,b,c...l) = n/2(a. &. c ... I) ?PD{0, a,b,c... I)}, 

^^.PD (0, a,b,c...l) = ^ {Jr{a, b,c...l) ^PD (0, a,b,c... I)}. 

This proposition admits of a great generalization*, but we have now all that 
is requisite for enabling us to arrive at a proposition exhibiting under one 
coiqy d'ceil every combination and every effect of every combination that can 
possibly be made with any number of coexisting equations of the first degree, 
containing any number of repeated, or to use the ordinary language of 
analysts, (variable or) unknown quantities. 

* See the Postscript to this paper for one specimen. 



50 On Derivation of Coexistence. [8 

For the sake of symmetry I make every equation homogeneous; so that 
to eliminate n repeated terms, no more than n equations will be required. 

In like manner the problem of determining n quantities from n equations 
will be here represented by the case in which we have to determine the 
ratios of (n + 1) quantities from n equations. 

Art. (9). Statement of the Equations of Coexistence. 

Let there be any number of bases {a, b, c ... I), and as many repeated 
terms (x, y, z ... t), and let the number of equations be any whatever, sa}' n. 
The system may be represented by the type equation 

arX + hry + CyZ + . • • +lrt=0, 

in which r can take up all integer values from — oo to + oo . The specific 
number of equations given will be represented by making the arguments of 
each base periodic, so that 

fi being any integer whatever. 

Art. (10). Combination of the given Equations. — Leading Theorem. 

Take f, g, ... k as the arbitrary bases of new and absolutely independent 
but periodic arguments, having the same index of periodicity (w) as a,b,c...l, 
and being in number {n — 1), that is, one fewer than there are units in that 
index. 

The number of differing arbitrary constants thus manufactured is 
n (n — 1 ). 

Let Ax + By+Cz+...+Lt = 0'be the general prime derivative from the 
given equations, then we may make 

A = ^PI)iO,aJ,g...k), 
B = ^PD{0,b,fg...k), 
C=^PD{0,c,fg...k), 



L^^PDiO,l,fg...k). 
Art. (11). CoE. 1. Inferences from the Leading Theorem. 

Let the number of equations, or, which is the same thing, the index of 
periodicity (w), be the same as the number of repeated terms (x, y, z ... t), 
then one relation exists between the coefficients: this is found by making 
the (n — 1) new bases coincide with (re — 1) out of the old bases. We get 
accordingly, as the result of elimination, 

fPi)(0, a, 6, c...0 = 0. 



8] On Derivation of Coexistence. 51 

Art. (11). CoE. 2. Let the number of equations be one more than that 
of the given bases, there will then be two equations of condition. These 
are represented by preserving one new arbitrary base, as \. The result of 
elimination being in this case 

CPi)(0, a, 6, C...Z, X)=0. 

Example. The result of eliminating between 
a-^x + hiy = 0, 
a^x + h^y = 0, 
a^x + h^y = 0, 
is fPX* (0, a, h, X) = 0, that is 

Xsb^ai — Xs^iCin + Xi^stto — Xibotts + Xgijas — XaftjCii = 0, 
from which we infer, seeing that Xg, Xj, Xj are independent, 
62 «! — ^i^a = 0, 
iatto — 621X3 = 0, 

any two of which imply the third. 

In like manner, in general, if the number of equations exceed in any 
manner the number of bases or repeated terms, the rule is to introduce so 
many new and arbitrary bases as together with the old bases shall make up 
the number of equations, and then equate the zeta-ic product of the differ- 
ences of zero, the old bases and the new bases, to nothing. 

Art. (12). Cor. 3. Let the number of equations be one fewer than the 
number (n) of bases or repeated terms ; the number of introduced bases in 
the general theorem is here (?; — 2). Make these (n— 2) bases equal severally 
to the bases which in the type equation are afBxed to z,u ...t, then 

D = 0, 

L = 0, 
and we have left simply 

^PD(0,a,c, d...kl)x + ^PD{0,b, c, d ...kl)y = 0. 
In like manner we may make to vanish all but A and C, and thus get 

?Pi)(0, a, b, d...U)x+^PD (0, c,b, d ... M) z=0, 

4—2 



52 



On Derivation of Coexistence. 



[8 



and similarly 



Hence 



fPD (0, a, 6 ... k) X + ^PD{0, h,c...l)t = Q. 
x\ i^PD{0,b,c...l) 

^PDia, 0,c...l) 
^PD{a,b, 0...1) 



are severally as <( 



tj \^PP{a,b,c...O). 

This is the symbolical representation as a formula of the remarkable 
method discovered by Cramer, perfected by Bezout and demonstrated by 
Laplace for the solution of simultaneous simple equations. 

Art. (13). Cor. 4. In like manner if the number of repeated terms 
be two greater than the number of equations, we have for the relation 
between any three of them, taken at pleasure, for instance, x, y, z, 

t,PP (0, a,d...l)x-\- i;PD (0, b,d...l)y+ ^PD (0, c, d ... I) z = 0. 

And in like manner we may proceed, however much in excess the number 
of repeated terms (unknown quantities) is over the number of equations. 

Art. (14). Subcorollary to Corollary 3. 

If there be any number of bases (a, b, c ...I), and any other two fewer in 
number {/, g ...k) 

^PD(a,f,g...k)x ^PD{b,c...l) 
+ ^PD{b,f, g ... k) X ^PD{a, c ... I) 
+ ^PD{c,f,g...k)x ^PD{b,c...l) 



+ tPD {l,f, g ...k)x ^PD (a,b,c... ) = 0. 

a formula that from its very nature suggests and proves a wide extension 
of itself. 

In conclusion I feel myself bound to state that the principal substance 
of Corollaries (1), (2) and (3) may be found in Garnier's Analyse Algebrique, 
in the chapter headed "Dev^loppement de la Th^orie donnee par M. Laplace, 
&c." But I am not aware of having been anticipated either in the fertile 
notation which serves to express them nor in the general theorems to which 
it has given birth. 

P.S. I shall content myself for the present with barely enunciating 
a theorem, one of a class destined it seems to the author to pla}^ no secondary 
part in the development of some of the most curious and interesting points 
of analysis. 

* The cross is used to denote ordinai-y algebraical multiplication. 



8] On Derivation of Coexistence. 53 

Let there be {n — 1) bases a, h, c ... I, and let the arguments of each be 
" recurrents of the ?ith order*," that is to say let 

27rA , , / SttA / 27rt\ , f Sttj 
. /). = i!r p.os . n. = A/ cos — 



dj = <^ ( COS ) , 6^ = t/t cos — j , c, = x\ cos — j , /i = ft) ( cos 

Let i?,. denote that any symmetrical function of the rth degree is to be 
taken of the quantities in a parenthesis which come after it, and let ^ 
indicate any function whatever. Then the zeta-ic product 

f [l;Rr {a, b,c...l)x ^^'^PD (0, a,b,c... I)} 

is equal to the product of the number 

itrl cos !-*/(— 1) sin — , cos |-v(-l)sin — , cos |-\/(-l)sm — , 

(V n n J \ n n J \ n n I 

2(?i-l)7r ,, ,, . 2(?i-l)7r' 
cos-^ ^+V(-l)sm ^ '— 



multiplied by the zeta-ic phase 

^p_^&Pi)(0, a, 6, C...0!! 

* I am indebted for this term to Professor De Morgan, whose pupil I may boast to have been. 
I have the sanction also of his authority, and that of another profound analyst, my colleague 
Mr Graves, for the use of the arbitrary terms zeta-ic, zeta-ically. I take this opportunity of 
retracting the symbol SPD used in my last paper, the letter S having no meaning except for 
English readers. I substitute for it QDP, where Q represents the Latin word Quadratus. On 
some future occasion I shall enlarge upon a new method of notation, whereby the language of 
analysis may be rendered much more expressive, depending essentially upon the use of similar 
figures inserted within one another, and containing numbers or letters, according as quantities 
or operations are to be denoted. This system to be carried out would require special but very 
simple printing types to be founded for the purpose. 

In the next part of this paper an easy and symmetrical mode will be given of representing any 
polynomial either in its developable or expanded form. 



A METHOD OF DETERMINING BY MERE INSPECTION THE 
DERIVATIVES FROM TWO EQUATIONS OF ANY DEGREE. 

[Philosophical Magazine, xvx. (1840), pp. 132 — 13.5.] 

Let there be two equations, one of the reth, the other of the mth degree 
in x; let the coefficients of the first equation be a^, a„_i, ffln-a ••• ctoi each 
power of X having a coefficient attached to it, a,j belonging to «" and a,, to 
the constant term. 

In like manner let hm, &m-i ... 6o be the coefficients of the second equation. 

I begin with 

A Rule for absolutely eliminating x. 

Form out of the (a) progression of coefficients m lines, and in like 
manner out of the (h) progression of coefficients form n lines in the follow- 
ing manner : 

1. (a) Attach {m — 1) zeros all to the right of the terms in the 
(a) progression ; next attach (?7i — 2) zeros to the right and carry over to the 
left ; next attach (m — 3) zeros to the right and carry over 2 to the left. 
Proceed in like manner until all the (m — 1) zeros are carried over to the 
left and none remain on the right. 

The m lines thus formed are to be written under one another. 

1. (6) Proceed in like manner to form n lines out of the (6) progression 
by scattering (m — 1) zeros between the right and left. 

2. If we write these n lines under the m lines last obtained, we shall 
have a solid square (m + n) terms deep and (m + n) terms broad. 

3. Denote the lines of this square by arbitrary characters, which write 
down in vertical order and permute in every possible way, but separate the 
permutations that can be derived from one another by an even number of 
interchanges (effected between contiguoxis terms) from the rest ; there will 
thus be half of one kind and half of another. 



9] On Elimination and Derivation hy mere Inspection. 55 

4. Now arrange the (m + n) lines accordingly, so as to obtain 

I {{in + ?i) (m + ?i - 1) . . . 2 . 1} 

squares of one kind which shall be called positive squares, and an equal 
number of the opposite kind which shall be called negative. 

Draw diagonals in the same direction in all the squares ; multiply the 
coefficients that stand in any diagonal line together : take the sum of the 
diagonal products of the positive squares, and the sum of the diagonal 
products of the negative squares; the difference between these two sums 
is the prime derivative of the zero degree, that is, is the result of elimination 
between the two given equations reduced to its ultimate state of simplicity, 
there will be no irrelevant factors to reject, and no terms which mutually 
destroy. 

Excmvple. To eliminate between 

ac(? + 6a; + c = 0, 
Ix^ + mx + n = 0, 



I write down 



a, b, c, 0, 

0, a, b, G, 

1, m, n, 0, 
0, I, m, n. 



(1) 

(2) 
(3) 
(4) 



I permute the four characters (1), (2), (3), (4), distinguishing them into 
positive and negative ; thus I write together 

Positive Permutations. 



1 


2 


3 


1 


2 


3 


2 


1 


3 


4 


4 


4 


2 


3 


1 


4 


4 


4 


1 


3 


2 


2 


1 


3 


3 


1 


2 


2 


3 


1 


4 


4 


4 


1 


3 


2 


4 


4 


4 


3 


1 


2 


3 


2 


1 


3 


2 


1 



and again 



Negative Permutations. 



1 


2 


3 


4 


4 


4 


2 


1 


3 


2 


1 


3 


2 


3 


1 


1 


2 


3 


4 


4 


4 


1 


3 


2 


4 


4 


4 


2 


3 


1 


1 


3 


2 


3 


2 


1 


3 


1 


2 


3 


1 


2 


3 


2 


1 


4 


4 


4 



66 



On Elimination and Derivation 



[9 



I reject from the permutations of each species all those where 1 or 3 
appear in the fourth place, and also those where 2 or 4 appear in the first 
place, for these will be presently seen to give rise to diagonal products 
which are zero. 

The permutations remaining are 

Positive effectual permutations. 



1 


3 


3 


1 


2 


1 


4 


3 


3 


2 


1 


4 


4 


4 


2 


2 



Negative effectual permutations. 



3 


1 


1 


3 


1 


4 


3 


2 


4 


3 


2 


1 


2 


2 


4 


4 



I now accordingly form four positive squares, which are 

a, b, c, 0, I, m, n, 0, I, m, n, 0, a, h, c, 0, 

0, a, b, c, a, b, c, 0, 0, I, m, n, I, m, n, 0, 

1, m, 11, 0, 0, a, b, c, a, b, c, 0, 0, I, m, n, 

0, I, m, n, 0, I, m, n, 0, a, b, c, 0, a, b, c. 

Drawing diagonal lines from left to right, and taking the sum of the 
diagonal products, I obtain a-n^ + Ib'n + Pc- + am-c. Again, the four negative 
squares 

1, m, n, 0, a, b, c, 0, a, b, c, 0, I, m, n, 0, 
a, b, 0, 0, 0, /, m, n, I, m, n, 0, 0, a, b, c, 
0, I, m, n, I, m, n, 0, 0, a, b, c, a, b, c, 0, 
0, a, b, c, 0, a, b, c, 0, I, in, n, 0, I, m, n, 

give as the sum of the diagonal products 

Ibmc + alnc + ambn + lacn, 

that is, Ibmc + ambn + 2acln. 

Thus the result of eliminating between 

ax^ + bx + c = 0, 

la? + mx + n = 0, 
ought to be, and is 

aV + Pc^ — 2acln + Ib-n + am^c — Ibmc — ambn — 0. 



9] hy a Process of mere Inspection. 57 



Kule for finding the prime derivative of the first degree, which is 
of the form Ax — B. 

Begin as before, only attach one zero less to each progression ; we 
shall thus obtain not a square, but an oblong broader than it is deep, con- 
taining (ni + n— 2) rows, and (m + w— 1) terms in each row: in a word, 
{m + ?i — 2) rows, and {m + n — \) cohimns. 

To find A reject the column at the extreme right, we thus recover 
a square arrangement (wi + /i — 2) terms broad and deep. 

Proceed with this new square as with the former one ; the difference 
between the sums of the positive and negative diagonal products will give A. 

To find B, do just the same thing, with the exception of striking off not 
the last column, but the last but one. 

Rxde for finding the prime derivative of any degree, say the rth, namely, 

ArX^ — Ar-iX'—^ + + ^0- 

Begin with adding zeros as before, but the number to be added to the 
(a) progression is (m — r) and to the (b) progression (n — r). 

There will thus be formed an oblong containing (m + n— 2r) rows, and 
(m + n — r) terms in each row, and therefore the same number of columns. 

To find any coefficient as Ag, strike off all the last (r+ 1) columns except 
that which is (s) places distant from the extreme right, and proceed with the 
resulting squares as before. 



Through the well-known ingenuity and kindly preferred help of a dis- 
tinguished friend, I trust to be able to get a machine made for working 
Sturm's theorem, and indeed all problems of derivation, after the method 
here expounded ; on which subject I have a great deal more yet to say, than 
can be inferred from this or my preceding papers. 



10. 

NOTE ON ELIMINATION. 

[Philosophical Magazine, XVII. (1840), pp. 379, 380.] 

The object of this brief note is to generalise Theorem 2 in my paper on 
Elimination* which appeared in the last December number of this Magazine. 
The theorem so generalised presents a symmetry which before was wanting. 
Here, as in so many other instances, the whole occupies in the memory a 
less space than the part. 

To avoid the ill-looking and slippery negative symbols, I warn my reader 
that I now use two rows of quantities written one over the other, to denote 
the product of the terms resulting from taking aiuay each quantity in the 
under from each in the upper row. 

Let Ai, Aj ... Am be the roots of one equation of coexistence, 

k-i, Ajj . . . kn of the other, 

and let the prime derivative of the degree r be required. Take any two 
integers p and q, such that p + q = r. The derivative in question may be 



written 



Ai/io ...hp\ [hp+Jip+. ...km 

'^q+l'^q+2 • ■ • "'71 



/ftl/lo... 

*■ \/lp+i/l^+2 ••• "m/ \"'g+i"'9+2 ••• "'«■ 

N.B. Whatever p and q be taken, so long only as p + q=^r, the above 
expression changes nothing but its sign ; which, therefore, upon transcendental 
grounds, it is easy to see is of one name or another, according as p is odd 
or even. 

In the original paper, I asserted this theorem only for the case oi p = 0, 
or 2=0. 

[* p. 43 above. En.] 



11. 



ON THE RELATION OF STURM'S AUXILIARY FUNCTIONS TO 
THE ROOTS OF AN ALGEBRAIC EQUATION. 

[Plymouth British Association Report 1841, (Pt II.), pp. 23, 24.] 

The author availed himself of the present meeting of the British 
Association to bring under the more general notice of mathematicians his 
discovery, made in the year 18.39, of the real nature and constitution of the 
auxiliary functions (so-called) which Sturm makes use of in locating the roots 
of an equation : these are obtained by proceeding with the left-hand side of 
the equation and its first differential coefficient as if it were our object tO' 
obtain their greatest common factor ; the successive remainders, with their 
signs alternately changed and preserved, constitute the functions in question. 
Each of these may be put under the form of a fraction, the denominator of 
which is a perfect square, or in fact the product of many : likewise the 
numerator contains a huge heap of factors of a similar form. 

These therefore, as well as the denominator, since they cannot influence 
the series of signs, may be rejected ; and furthermore we may, if we please, 
again make every other function, beginning from the last but one, change its 
sign, if we consent to use changes wherever Sturm speaks of continuations 
of sign, and vice versa. 

The functions of Sturm, thus modified and purged of irrelevancy, the 
author, by way of distinction, and still to attribute honour where it is really 
most due, proposes to call " Sturm's Determinators " ; and he proceeds to lay 
bare the internal anatomy of these remarkable forms. 

He uses the Greek letter " ^" to indicate that the squared product of the 
differences of the letters before which it is prefixed is to be taken. 

Let the roots of the equation be called respectively a, b, c, e...l, the deter- 
minators taken in the inverse order are as follows : — 

^(a, b, c, e ... I). 

X^{b, c, e ...I) X -ta^ib, c, e ...I). 

2^(c, e...l)ay'-X{a + b).^(c,e...l)x + tab.^{c, e...l). 

******* 
2 {^{k, I) {x -a)(x-b){x-c)ix-e)...{x- h)}. 



60 Sturm's Auxiliary Functions. [11 

It may be here remarked, that the work of assigning the total number of 
real and of imaginary roots falls exclusively upon the coefficients of the 
leading terms, which the author proposes to call " Sturm's Superiors " : these 
superiors are only partial symmetric functions of the squared differences, but 
complete symmetric functions of the roots themselves, differing in the former 
respect from those other (at first sight similar-looking) functions of the 
squared differences of the roots, in which, from the time of Waring downwards, 
the conditions of reality have been sought for. It seems to have escaped 
observation, that the series of terms constituting any one of the coefficients 
in the equation of the squares of the differences (with the exception of the 
first and last) each admit of being separated and classified into various 
subordinate groups in such a way, that instead of being treated as a single 
symmetric function of the 7-oots, they ought to be viewed as aggregates of 
many. In fact, Sturm's superior No. 1 is identical with Waring's coefficient 
No. 1 ; Sturm's superior No. 2 is a part of Waring's coefficient No. 3 ; Sturm's 
superior No. 3 is a part of Waring's coefficient No. 6 ," and so forth till we 
come to Sturm's final superior, which is again coextensive and identical with 
the last coefficient in the equation of the squares of the differences. The 
theory of symmetric functions of forms which are themselves symmetric 
functions of simple letters, or even of other forms, the author states his belief 
is here for the first time shadowed forth, but would be beside his present 
object to enter further into. He would conclude by calling attention to the 
importance to the general interests of algebraical and arithmetical science 
that a searching investigation should be instituted for showing, a priori, how, 
when a set of quantities is known to be made up partly of possible and partly 
of pairs of impossible values, symmetrical functions of these, one less in 
number than the quantities themselves, may be formed, from the signs of the 
ratios of which to unity and to one another the respective amounts of possible 
and impossible quantities may at once be inferred : in short, we ought not to 
rest satisfied, until, from the very form, of Sturm's Determinators, without 
earing to know how they have been obtained, we are able to pronounce upon 
the uses to which they may be applied. 



12. 



EXAMPLES OF THE DIALYTIC METHOD OF ELIMINATION 
AS APPLIED TO TERNARY SYSTEMS OF EQUATIONS. 

[Cambridge Mathematical Journal, ii. (1841), pp. 232 — 236.] 

This method is of universal application, and at once enables us to reduce 
any case of elimination to the form of a problem, where that operation is to 
be effected, between quantities linearly involved in the equations which 
contain them. 

As applied to a binary system, fx =0, ^« = 0, the method furnishes 
a rule by which we ma}' unfailingly arrive at the determinant, free from every 
species of irrelevancy, whether of a linear, factorial, or numerical kind. 

The rule itself is given in the Philosophical Magazine (London and 
Edinburgh, Dec. 1840). The principle of the rule will be found correctly 
stated by Professor Richelot, of Konigsberg, in a late number of Crelle's 
Journal, at the commencement of a memoir in Latin bordering on the same 
subject (" Nota ad Eliminationem pertinens "). 

My object at present is to supply a few instances of its application to 
ternary systems of equations. 

Ex. 1. To eliminate x, y, z, between the three homogeneous equations 
Af--Wxy + Bx^ = Q, (1) 

Bz^ - 2A'yz + Cif = 0, (2) 

Gx- - 2B'zx + Az- = 0. (3) 

Multiply the equations in order by —z-, x", y'', add together, and divide 
out by 2xy ; we obtain 

C'z^ + Gxy — A'xz- B'yz = 0. (4) 

By similar processes we obtain 

A'x- + Ayz — B'yx - C'zx = 0, (5) 

B'y"- + Bzx - G'zy - A'xy = 0. (6) 



62 Examples in Dialytic Elimination. [12 

Between these six, treated as simple equations, the six functions of 
■X, y, z, namely, x^, y", z-, xy, xz, yz, treated as independent of each other, may 
be eliminated ; the results may be seen, by mere inspection, to come out 

ABC {ABC - AB'^ - BC - GA'^- + 2A'B'G') = 0, 

or rejecting the special (N.B. not irrelevant) factor ABC, we obtain 

ABC -AB'- BG" -CA'' + 2A'B'C' = 0. 

I may remark, that the equations (1), (2), (3), or (4), (5), (6), express the 
condition of 

Ax- + By^ + Cz^ + 2A'yz+2B'zx + 2G'xy, 

baving a factor Xx + fj,y + vz ; a general symbolical formula of which I am in 
possession for determining in general the condition of any polynomial of 
any degree having a factor, furnishes me at once with either of the two 
systems indifferently. The aversion I felt to reject either, led me to employ 
l)oth, and thus was the occasion of the Dialytic Principle of Solution mani- 
festing itself, 

Ex. 2. Ax- + ayz + hzx + cxy'= 0, (1) 

My- + lyz + mzx + nxy = 0, (2) 

Rz- + pi/z + qzx + rxy = 0. (3) 

Multiply equation (1) by ^y+yz, equations (2) and (3) by vz and Ky 
respectively, and add the products together, we obtain terms of which y^z 
and yz^ are the only two into which x does not enter. 

Make now the coeflEicients of each of these zero, and we have 
ay + lv + Rk = 0, 
aj3 + Mv-\-pK = Q. 
Let v = a, K = a, then y = -{l + R), ^ = -(ilf + jj). 
Hence, multiplying as directed, and then dividing out by x, we obtain 

(mv 4- by) z^ + (tk + C0) y- + (6/S -\-cy + nv + qx) yz + A^xy + Ayxz = 0, 
or by substitution, 

[ra - c {M + p)} y^ + {ma-b{l + R)} z" + {an + ag - 6 (ilf + p) - c (^ + R)] yz 
-A{M+p)xy-A{M.^-p)xz = 0. (4) 

Similarly, by preparing the equations so as to admit in turn of y and z 
as a divisor, we obtain 
[ma -1{R + b)} z"- + {mr — n{A + q)} x" + {mc + mp — n {R ^-b) -I {A -^ y)} xz 

-M(R + b)yz-A(A+q)xy=0, (5) 

[rm - q{A + n)] x' + {ra — p{M + c)} y^ +{rl + rb-'p{A + n) -q{M + c)} xy 
-R(A+n)xz-R(M+c)yz=0. (6) 



12] Examples in Dialytic Elimination. 63 

Between the six equations (1), (2), (3), (4), (5), (6), x^, y-, z^, xy, xz, yz, may- 
be eliminated ; the result will be a function of nine letters {three out of each 
equation (1), (2), (3)j equated to zero. Perhaps the determinant may be 
found to contain a special factor of three letters ; and if so, may be replaced 
by a simpler function of six letters only. 

Ex. 3. To eliminate between the three general equations 
Ax- + By- + Gz'- + 2Dyz + 2Ezx + 2Fxy = 0, 
Lx' + My"- + Nz"- + 2Pyz + ^Qzx + 2Rxy = 0, 
fx + gy + hz=0. 

By virtue of one of the two canons which limit the forms in which the 
letters can appear combined in the determinant of a general system of 
equations, we know that the determinant in this ' case (freed of irrelevant 
factors) ought to be made up in every term of eight letters (powers being 
counted as repetitions), namely, (A , B, C, D, E, F) must enter in binary com- 
binations, (Z, M, N, P, Q, R) the same, whereas/, g, h must enter in quaternary 
combinations. 

To obtain the determinant, write 

Ax- + By^ + Cz^ + Dyz + Ezx + Fxy = 0, (1) 

Lx'' + My-+Nz-+ Pyz + Qzx + Rxy = 0, (2) 

fx"" + gyx + hzx = 0, (3) 

fxy+gy'' + hzy =0, (4) 

fxz + gyz + hz^ = 0. (5) 

We want one equation more of three letters between «^ y^, z^, xy, xz, yz. 
To obtain this, write 

{Ax + Ez + Fy) «! + {By + Fx + Dz) y^ + {Cz -{■I>y + Ex) z^ = 0, 

{Lx -\-Qz-\- By) x^ + {My ^ Bx + Pz) y^ + {Nz + Py-^ Qx) z-, = 0, 

/«i + gyi + M = 0. 

Forget that x-^ =x, y^ = y, ^^ = z, and eliminate Xj, y^, z^, we obtain 

, ({Ax + Ez+ Fy) {My + Rx + Pz) 
j- {By + Fx + Dz) {Lx -i-Qz + Ry) 

{{Cz + Dy + Ex){Lx+Qz + Ry)\ 
^\-{Nz + Py+Qx){Ax+Ez + Fy)] 

({Nz + Py + Qx){By + Fx + Dz)\_ 
^■^\-{Oz + Dy + Ex){My + Rx + Pz)) 



64 Examples in Dialytic Elimination. [12 

This may be put under the form 

ax^ + ^y- + 72- + a'yz + jS'zx + y'xy = 0, (6) 

where the coefficients are of the first order in respect to f, g, h, L, M, N, 
P, Q, R, A, B, C, D,E,F; in all of the third order. 

Between the equations marked from (1) to (6), the pirocess of linear 
elimination being gone through, we obtain as equated to zero a function of 
5 + 3, or of eight letters, two belonging to the fii-st equation, two to the 
second, and four to the third ; so that the determinant is clear of all factorial 
irrelevancy. 

Ex. 4. To eliminate x, y, z between the three equations 
Ax" + By-" + Cz-" + lA'yz + IB'zx + IG'xy = 0, 
Lap- + My- + Nz" + IL'yz + IM'zx + IN'xy = 0, 
P«2 + Q/ + Uz" + IFyz + iqzx + lExy = 0. 
Call these three equations U=0, V=0, W =0, respectively. Write 
xU=0, (1) yll^O, (2) zU=0, (3) 

xr=0, (4) yV=0, (5) zV=0, (6) 

xW=0, (1) yW=0, (8) zW=0. (9) 

We have here nine unilateral equations : one more is wanted to enable us 
to eliminate linearly the ten quantities 

a;^, y^, z^, x'y, x^z, xy"-, xz"^, xyz, y'^z, yz"^. 

This tenth may be found by eliminating x, y, z between the three equations 

X {Ax + B'z + G'y) -\-y{By+ C'x + A'z) + z (Cz + A'y + B'x) = 0, 

X (Lx + M'z + N'y) + y {My + N'x + L'z) + z {Nz + L'y + M'x) = 0, 

X {Px + Q'z + Ry) + y{Qy + B'x + P'z) + z{Rz + P'y + Q'x) = 0; 

for, by forgetting the relations between the bracketed and unbracketed letters, 
we obtain 

{Ax^Bz + G'A {My + N'x + L'z){Rz + P'y + qx)) 

+ &c. + &c. = 0, 
which may be put under the form 

ax'' + ^tf + yz^ + hx-y + =0*. (10) 

* We might dispense with a 10th equation, using the nine above given, to determine the 
ratios of the ten quantities involved to one another ; and then by means of any such relations as 

x^yxxy'^ = x^y-y.x-y", or x^xy^=x-y xxy', &c. 
obtain a determinant. But it is easy to see that this would be made up of terms, each containing 
literal combinations of the 18th order. 

Again, we might use five out of the nine equations to obtain a new equation free from 
!/') y'^^i y^'i ^' ; ^^^^ i^i containing x in every term : which being divided by x, and multiplied 



12] Examples in Dialytic Elimination. 65 

By eliminating linearly between the equations marked from (1) to (10), 
we obtain as zero a quantity of the twelfth order in all, being of the fourth 
order in respect to the coefficients of each of the three equations, which is 
therefore the determinant in its simplest form. 

I have purposely, in this brief paper, avoided discussing any theoretical 
question. I may take some other opportunity of enlarging upon several 
points which have hitherto been little considered in the theory of elimination, 
such as the Canons of Form, — the Doctrine of Special Factors, — the Method 
of Multipliers as extended to a system of any order, — the Connexion between 
the method of Multipliers and the Dialytic Process, — the Idea of Derivations: 
and of Prime Derivatives extended to ultra-binary Systems. For the present 
I conclude with the expression of my best wishes for the continued success of 
this valuable Journal. 

by y, or by z, would furnish a 10th equation no longer linearly involved in the 9 already found. 
The determinant, however, found in this way, would consist of 14-ary combinations of letters. 

Finally, we might, instead of a system of ten equations, employ a system of 15, obtained by 
multiplying each of the given three by any 5 out of the 6 quantities x^, y^, z^, xy, xz, yz ; but the 
determinant, besides being not totally symmetrical, would contain combinations of the 15th 
order. 

I may take this opportunity of just adverting to the fact, that the method in the text does in 
fact contain a solution of the equation 

•\i7 + ,u.F+ vW=x^y'>z', 

where r + s + t = i, and \, /j., v are functions of the second degree in regard to x, y, z to be 
determined. 



13. 



INTRODUCTION TO AN ESSAY ON THE AMOUNT AND DIS- 
TRIBUTION OF THE MULTIPLICITY OF THE ROOTS OF 
AN ALGEBRAIC EQUATION. 

[Philosophical Magazine, xviii. (1841), pp. 136 — 139.] 

I USE the word multiplicity to denote a number, and distinguish between 
the total and partial multiplicities of the roots of an algebraic equation. 

There may be r different roots repeated respectively A, , h^ ... hr times. 
r is the index of distribution. 

/(t , ho... h,. are the partial multiplicities, and if h = h^ + /1.2 + . . . + hr 
h is the total multiplicity. 

The total multiplicity it is clear may be defined as the difference between 
the index of the equation and the number of its roots distinguishable from 
one another. 

In this Introduction, I propose merely to consider how existing methods 
may be applied to determine the amount and distribution of multiplicity 
in a given equation, and conversely, how equations of condition can be 
formed which shall imply a given distribution and amount. 

Let the greatest common factor between fx (the argument of the pro- 

dfx 
posed equation) and -^ be called f^x. 

And in like manner, let the greatest common factor oi fiX and —^ be 
called f^x and so on, till in the end we come to frX, which has no common 

factor with -^^ , 
ax 

Let ki, k„...kr denote the degrees in x oifx,fiX ...frX respectively. 

It is easy to see that 

^1 — ^2, partial multiplicities, are less than 2, that is, are each units. 

^2 — ^'3, partial multiplicities, will be less than 3, and therefore either 1 
or 2 in value respectively, and so on till we come to 

kr-i — kr which will severally be between zero and r — 1, and 

kr — of values intermediate between zero and r. 



13] On the Multiplicity of an Algebraic Equation. 67 

Hence there will be 

ki — ^k^ + ks multiplicities each of the value 1, 
h - 2^3 + h „ „ „ 2, 

kr-i — 2kr ... of the value r—1, 

and kr of the value r. 

In place of fx with -~- we might employ -4— with -,— and so on for 
^ •' dx ° '^ •' dx dx^ 

the rest; the values of k^tk^.-.k^ will remain unafifected by this change; 
but the former method would be more expeditious in practice. 
The total multiplicity is, of course, =ki. 

Suppose now that we propose to ourselves the converse problem to 
determine the conditions that an algebraic equation may have a given 
amount of multiplicity distributed in a given manner. 

I{ hi, h.2, h^... hr be used to denote the given number of partial multi- 
plicities which are respectively of the values 1 , 2, 3 ... r, it is easy to see 
that the quantities derived above by k^, k^.-.kr are respectively equal to 

}h + 2h^+ +rhr, 

L + 2h3+ +rhr-i, 

h^ + 2h,+ +r/;^_2, 

K. 

dfx 
Now from -j— having a factor of the degree ki common with fx we obtain 

dfx 
ki conditions, from -^^ having a factor of the degree ^2 common withy, a; we 

obtain k„ more, and so on. So that altogether we obtain in this way 
ki + k3+ +kr conditions. 

But it may easily be seen that the total multiplicity being k^, the number 
of conditions need never to exceed k^ in number, no matter what its distri- 
bution may be. Hence, besides the enormous labour of the process, and the 
extreme complexity of the results, we obtain by this method more equations 
by far than are necessary, and it requires some caution to know which to 
reject. 

In my forthcoming paper (to appear in Philosophical Magazine of next 
month) I shall show, by a most simple means, how without the use of derived or 
other subsidiary functions, to obta,in the simplest equations of condition which 
correspond to a given distribution of a given amount of multiplicity. 

The total multiplicity, say m, being given in as many ways as that 
number can be broken into parts, so many different systems of in equations 
can be formed differing each from the other in the dimensions of the terms. 

5—2 



68 On the Multiplicity of an Algebraic Equation. [13 

These systems may be arranged in order so that each in the series shall 
imply all those that follow it, and be implied in all those that go before, 
without the converse being satisfied. 

The subject of the unreciprocal implication of systems of equations is 
a very curious one, upon which the limits assigned to me prevent me from 
enlarging at present. It is closely connected with a part of the theory of 
elimination, which, as far as I am aware, has either been overlooked, or has 
not met with the attention which it deserves ; I mean the theory of Special 
Factors. 

An example may make what I mean by these clear. 

Let be a function (if my reader please) void of x, which equivalent to 
zero implies two given equations in x having a common root. 

Let G be rid of all irrelevant factors, that is, let C be the simplest form 
of the determinant, when the coefficients of the two equations are perfectly 
independent qualities. Now suppose, as is quite possible in a variety of ivays, 
that such relations are instituted between the coefficients alluded to as make 
C split up into factors, so that G = L x M y. N =0. 

Only one of the factors L, M, N will satisfy the condition of the co- 
existence of the two given equations : the others are clearly, however, not to 
be confounded with factors of solution, or irrelevant factors, as they are 
termed, but ai-e of quite a different nature, and enjoy remarkable properties, 
which point to an enlarged theory of elimination, and constitute what I call 
special or singular factors. 

I shall feel much obliged to any of the readers of your widely circulated 
Journal, interested in the subject of this paper, who would do me the honour 
of communicating with me upon it, and especially if they would (between 
now and the next coming out of the Magazine) inform me whether any- 
thing, and if so how much, different from what is here stated has been done 
in the matter of determining the relations between the coefficients of an 
equation corresponding to a given amount and distribution of multiplicity in 
its roots. 

I ought to add, that my method enables me not merely to determine 
the conditions of multiplicity, but also to decompose the equations con- 
taining multiple roots into others free of multiplicity, that is, to find, 
a priori, the values of the several quantities 

if^xf if^xy ' (frxf^'- ■ 

Moreover, other decompositions, not necessary to be enlarged upon in this 
place, may be obtained with equal facility. 



14. 

A NEW AND MORE GENERAL THEORY OF MULTIPLE ROOTS. 

[Philosophical Magazine, XVIII. (1841), pp. 249 — 2.54.] 

I SHALL begin with developing the theory of polynomials containing 
perfect square factors, one or more. 

First, let us proceed to determine the relations which must exist between 
the coefficients of such polynomials, and afterwards show how they may be 
broken up into others of an inferior degree. 

A parallelogram filled with letters standing in one row is intended to 
express the product of the squared difference of the quantities contained. 
Thus (a6) indicates (a — &y, (^abc^ is used to indicate (a—by{a — cf{b — c)', 
and so forth. 

Suppose now that two of the roots gj, e, ... «„ belonging to the equation 
fx = are equal to one another, it is clear that (ei, e™ ... e„) = ; and more- 
over is a symmetric function, and can be calculated in terms of the coefficients 
oifx. 

Next let us suppose that we have two couples of equals (as for instance 
a and b, two of the roots equal, as also c and d two others), it is clear, that on 
leaving any one of the roots out, the (n — 1) that are left will still contain 
one equality, and therefore we have 

( bj, e3...e„ )=0, ( e,, e^ ... e„ ) = ... ( ei, e^ . . . e„_i ) = 0. 

None of the parallelogrammatic functions above taken singly, are symmetric 
functions of the coefficients, but their sum is ; so also is the sum of the 
product of each into the quantity left out. 

Now in general, suppose that the polynomial fx contains r perfect square 
factors, so that we have r couples of equal roots belonging to the equation 

fx = 0,it is clear that (e,-, e^+i ... e™) and all the other ., — „ ""; =- 

•' V ^ -^ 1.2..,(r — 1) 

functions of which it is the type are severally zero. Moreover, the sum of 



70 On a new and more general [14 

these or the sum of the products of each by any symmetrical function of the 
(r — 1) letters left out will be a symmetrical function of the coefficients of 
the powers of x in fx. To express now the affirmative* conditions corre- 
sponding to the case of there being r pairs of equal roots, we might employ 
the r equations, 



(e. 


e2...e«) = 


= 0, 


2(e^ 


ei...e„) = 


= 0, 


2(^ 


....„) = 


= 0, 








2(.. 


, e^+i ...e„) = 


= 0. 



But these, except the last, are not the simplest that can be employed ; that 
is to say, we can write down r others, the terms of which shall be of lower 
dimensions in respect to the roots. 

Let /^ denote that any rational symmetrical function of the /xth degree 
is to be taken of the quantities which it precedes. 

Then the r equations in question are all contained in the general equation 

S {/^ (ei, 62 . . . e,_i) X ( er,er+i...en )} = ; 

fi being taken from up to (r — 1) we obtain r equations, which in respect 
to the roots are respectively of all degi-ees between 

n(n-l)...(n-r + 2) n{n -1) ... {n-r + 2) 

1.2...(r-l) ^'"'^ 1.2...0--i) +^^--^^ 

reckoned inclusivelj\ 

Now at this stage it is important to remark that the above r equations, 
although necessary, are not sufficient; and indeed, no mere affirmations of 
equality can be sufficient to ensure there being r pairs of equal roots. 

To make this manifest, suppose r = 2. Then in order that an equation 
may have two pairs of equal roots, we must have by the above formula 

2 ( ea, 63 . . ■ en ) = 0, 2 {fii ( eg, 63 • ■ • e,, )} = 0. 

But if instead of there being two perfect square factors there be one 
perfect cube factor in/«, it may be shown by the same reasoning as above, that 
t he very same two equations apply. In fact, it may be shown in general 
that no such equations as those given above can be affirmed in consequence 
of there being an amount r of multiplicity consisting of unit parts which 
may not be affirmed with equal truth as necessary consequences of thb same 

* The importance of the restriction hinted at by the use of the word affirmative will appear 
hereafter. 



14] Theory of Multiple Roots. 71 

amount distributed in any other manner whatever. How to obtain affirma- 
tive equations sufficient as well as necessary (under certain limitations) will 
appear at the close of this present paper. 

It is worthy of being remarked, that if we make ^x denote the sum of 

the products of the quantities to which it is prefixed, taken fj, and /n together, 

the equations of affirmation become identical with those obtained by elimin- 

dfx 
atiug between /« and ~— *. 

It can scarcely be doubted that the illustrious Lagrange, had he chosen 
to perfect the incomplete theory of equal roots given in the Resolution 
Nmnerique, by applying to it his own favourite engine of symmetric func- 
tions, could scarcely have failed of stumbling by a back passage upon Sturm's 
memorable theorem. 

Let us now proceed to show how a polynomial known to contain one or 
more perfect square factors may be decomposed. 

Let us begin with supposing that it contains but one such factor; so 
tla.&tfx = cf)x(x — a)-. 

I shall show how to obtain the equations 

C(a;-a) = 0, Dcf,x(a; - a) = 0, E(x-ay = 0, F{4>x) = 0, 

each in its lowest terms. 

1. To form the equation Lx + M = 0, where a; = a, it is easy to see that 
if we write down in general the expression {x — e^) (en, e^ . . . en) this will 
become zero whenever the root e-i left out is not one of the equal roots (a) : 
so that in fact (calling the two equal roots Sj, e, respectively) 



2 {{x - ei) X ( e., 63... en )} = (x — e^ x (ca, e^ ... e„ ) +{x- e,) x ( ei, 63 ...en ), 

or simply =2(x — a) le^, e^ ... e^). 

Hence by making 

x1 {en, 63... en) — 2 {ei X (e^, e-,... en)} = 0, 
we have an equation for finding the equal roots e^, e^. 
Again, it is easily seen upon the same hypothesis, that 

2 {{x — 65) {x - 63) {x — e^...{x — en) x ( eg, 63 ... Bn) } 
= 2 (a; — 62) {x - 63) ...(« — e„) X ( eg, 63... e^ ). 

* See my note on Sturm's Theorem, Phil. Mag., December, 1839 [p. 45 above. Ed.J, 



72 On a new and more general [14 

Hence, to form the equatioa having the same roots as (« — a) (j)x, we have 
only to make 

a;"-^ S ( eg, 63 ■ ■ . Cn ) — a;"~' 2 {(fe + 63 + ... e„) x ( e^, 63 ■.. e„ )j 

+ 2 {(6263 . . . e„) X (e,, 63 . . . e,i)} = 0. 

Suppose now in general that we have r perfect square factors, so that 

fx = (^x{x — aiY {x — a^"^ ...{x — ar)". 

To form the equation (7(« — cij) (a?— aj) ... (a:— a,.) = 0, we have only to 
make 

2 {(x — Ci) (x — Bz) ... (X — Br) X (e^+i, 6^+2 ■ ■ . Sn)} = 0. 

And to obtain 

D(J3X X {x — ai) (x — a«) ... (x — a^) = 0, 
we must make 

2 {(« - e,.+i) (a; - er+2) ...(«- e„) x ( e^+i, e^+, ... e„ )} = 0. 

The theory of perfect square factors is not yet complete until it has been 
shown how to obtain constructively (px, and, as analogy suggests, the com- 
plementary part D' {x — aif {x — a^)- ...{x — a,)-, each in its lowest terms. 
To effect the latter it might be said that it is only necessary to take the 
square of (a; — a^) (x — ao) ...(x — a^). It is true the polynomial so formed 
would contain every pair of equal factors, but not in the lowest terms as 
regards the coefficients (as we shall presently show). 

To solve this last part of the problem, let it be agreed that two rows of 
letters inclosed in a parenthesis shall indicate the product of the squares 
of the differences got by subtracting each in the row from each in the other, 
so that 

= (a - h)\ (f'] = (a - by (a - c^, C"^) = (a - cf (a - dy (h - cf (b - df. 



bj~^"' "" \bc)'^"' ' ^ " \cd. 

Let us begin with supposing that fx has one pair only of equal roots ; 
to form the simplest quadratic equation containing this pair, write down 



{x - 6]) {x - e.) X (ea, 64 . . . e„) X 

^ \6i > 64 ... 6n. 

Now if 61 and e^ are the two equal roots in question neither of the 
multipliers of (x — 61) (x — e^) vanishes. 

If 61 and 62 are neither of them equal roots ^63, 64 ... e„) = 0. 
If one of the two only belong to the pair of equal roots 



14] Theory of Multixole Roots. 73 

Hence it is clear that 



S j (« — Ci) {x — e„) X ^63, 64 . . . e,i) X 



61 > 60 



eJl 



is the equation desired. 



In like manner if there be r pairs of equal roots the equation of the 
(2?')th degree which contains them all may be written 



( V finr+i ... Sn / I 

The coefficient of x-^ in this equation is clearly of 

{n - 2r) {n - 2r - 1) + 4?- {n - 2r), 

that is, o{ (n + 2r — l)(n — 2r) dimensions. The coefficient of x" in the equa- 
tion which contains the r equal roots unyoked together is of (n — r){n — r—l) 
dimensions, and consequently the coefficient of x"" in the square of this 
equation would be of 2(n —r) {n- r — 1) dimensions, that is, would be 
n-+ 6r'- — (4?- + 1) n dimensions higher than needful. 

Finally, to obtain an equation clear of simple as well as double appear- 
ances of the equal roots, we have only to write the complementary form 



S \(x — e^+i) (x - ea--(-2) ... (x- Bn) X ( e^r+i + en ) X ( " ''"' '^)^=0. 

Let us, now that we are more familiarized with the notation essential to 
this method, revert to the question with which we set out, and endeavour to 
obtain r such equations as shall imply unambiguously the existence of r pairs 
of equal roots. 

The existence of r such pairs enables us to assert the following disjunc- 
tive proposition, which cannot be asserted when the same amount of multi- 
plicity is distributed in any other way. 

To wit, on selecting any r roots out of the entire number, either these 
r will all be found again in those that are left, or those that are left will 
contain inter se, one repetition at least ; so that except on the latter supposi- 
tion any (i — 1) may be absolutely sunk out of those that are left, and there 
will still be one root common to the (n—2r+l) remaining, and to the r 
originally selected to be left out. 

Wherefore calling the roots ej, 62 •■• ^m and giving /a any value whatever, 
we have 

2 1 J^(.„ e. ... e.) X (^...e,...-e.) X 2 £;:;;;^: J} = o. 



74 Theory of Multiple Roots. [14 

Hence the simplest distinctive equations indicative of the existence of r 
pairs of equal roots are to be found by putting jj, equal in succession to all 
values from up to (r — 1). 

For instance, if we require that an equation of the seventh degree shall 
have three pairs of equal roots, we need only to call the seven roots respec- 
tively a, h, c, d, e,f, g, and then our type equation becomes 

.J/ ' V^^/ \ ^ ( ef\^ eg \^rtg\\ 



\ \a b cj ' \a b cj ' \a b C/ 

From this it appears that the r distinctive equations for r pairs of equal 
roots are of different dimensions from the r general or overlying ones corre- 
sponding to the multiples r, anyhow distributed; the lowest of the latter 
being oi (n — r +l){n — r), the lowest of the former of 

(n — r){n — r—l) + 2r {n — 2r -1- 1), 

that is, of n(n — l) — 3r (n — 1) dimensions. In general we shall find that 
the more unequally distributed the multiplicity may be the lower are the 
dimensions of the distinctive equations, and are accordingly lowest when the 
multiplicity is absolutely undistributed*. 

* It must not, however, be overlooked, that the equations above given, although decisive as 
to the existence of r pairs of equal roots when the multiplicity is known to be not greater than r, 
do not enable us to affirm with certainty their existence when this limitation is absent : for 
should the multiplicity exceed r, then inevitably (no matter how it may be distributed) 
(cr+i' ^r+2 ■•■ O is always zero, and consequently nullifies each term of every one of the equa- 
tions in question. In fact (repugnant as it may appear to be to the ordinary assumptions of 
analytical reasoning), it is not possible to express with absolute unambiguity the conditions of 
there being a multiplicity (r) distributed in any assigned manner by means of r affirvmtive 
eqtiations alone. 



15. 



ON A LINEAR METHOD OF ELIMINATING BETWEEN DOUBLE, 
TREBLE, AND OTHER SYSTEMS OF ALGEBRAIC EQUATIONS. 

[Philosophical Magazine, xviil. (1841), pp. 425 — 435.] 

Part I. Binary Systems. 

Let U and V be two integer complete homogeneous functions of x and 
y, one of the mth, the other of the wth degree ; and let it be required to 
express the condition of the coexistence of the two equations U=0, V =0 
by means of the equation 0=0, where C is free from all appearances of 
X or y. 

This equation, according to the system of notation developed in a pre- 
ceding paper, and which has been since adopted and sanctioned by the high 
authority of M. Cauchy, I call the final derivative : the quantity G is desig- 
nated the final derivee : and it is our present object to show how this may 
be obtained in a prime form, that is to say, divested of irrelevant factors : 
in this state it must consist of terms, each containing m + n letters, of which 
n belong to the coefficients of U, and m to those of V. 

Of course in applying this rule it is to be understood that every combina- 
tion of powers in IT or V has a single letter prefixed for its coefficient, 
and that in the final derivee powers are represented by repetitions of the 
same character. 

Every term in U or V being of the form CxPy'J, x^yi is called an argu- 
ment, G its prefix. 

Assume two integer positive numbers r and »-', and also two others 
s and s', such that r + r' = n — 1, s + s' = m — 1, and form from 11=0, V= 
two new equations, 

x'y''"U=0, afy'-'V=0. 

Such equations are termed the augmentatives of the two given ones respec- 
tively ; also x^y'^ U and its fellow are termed the augmentees of U and V. 



76 On a linear Method of Eliminating between [15 

r and r' are termed the indices of augmentation belonging to U, s and s' 
the same belonging to V. 

Finally, it will be useful hereafter to call the given polynomials TJ and V 
themselves the proposees, and the given equations which assert their nullity, 
the prepositive equations, or, briefly, the propositives. 

Now as many augmentees of either proposee can be formed as there are 
ways of stowing away between two lockers (vacancies admissible) a number 
of things equal to the index of the other* ; hence we shall have n aug- 
mentees of TJ, and m of F": thus there will be m + w augmentatives each of 
the degree m+n~\, and the number of arguments is clearly m + n also, 
so that they can be eliminated linearly, and the final derivee thus found, 
containing m + n letters (properly aggregated) in each term, will be in 
its prime form, that is, incapable of further reduction, and void of irrelevant 
factors. 

It is worthy of remark, that the final derivee obtained by arranging in 
square battalion the prefixes of the augmentees, permuting the rows or 
columns, and reading off diagonal products, affected each with the proper 
sign (according to the well known rule of Duality), will not only be free 
from factorial irrelevancy, but also of linear redundancy, which latter term 
I use to signify the reappearance of the same combination of prefixes, some- 
times with positive and sometimes with negative signs : furthermore, it 
follows obviously from the nature of the process that no numerical quantity 
in the final derivee will be greater than the higher of the indices of the two 
given polynomials. 



Part II. Ternary Systems. 

Case A. Indices all equal. 

Method 1. 

Let there be now three proposees, TJ, V, W, integer complete homo- 
geneous functions of x, y, z, each of the degree n : let 

r + 7-' + r"=n-l, s + s' + s" =n-l, t + t' + t" = n-l, 
x^y'' z'^'TJ, afy^'z^'V, x*y^'z*"W, 
will, as above, be called the augmentees of TJ, V, W, and every other part of 
the notation previously described is to be preserved. 

"Tot Augmeuta utriusvis ex squationibus propositis formari possunt quot modi sint inter 
duo receptaeula (utrivis vel ambobus omnino vaeare licet) rerum, quarum numerus indieem 
alterius squat, distributionem faciendi." 



15] double, treble, and other Systems of Algebraic Equations. 77 

Suppose now 

U=0, F=0, F = 0, 

we shall have as many augmentative equations formed from each proposee 
as there are ways of stowing away n things between three lockers (vacancies 

admissible)*, that is, n „ of each kind; in all, therefore, 3 — -, and 

every one of these will be of the degree 2)i — 1, so that the number of 
arguments to be eliminated is equal to the number of ways of stowing 
away 2n — 1 things between three lockers (empty ones counting), that is 

2w (2?i + 1) 
2 

As yet, then, we have not enough equations for eliminating these linearly. 

Make, however, 

and write 

U=x'-F+ %fF' + z-^F" = 0, 

W = af-H+ ym' + zyH"=- 0, 
it will always be possible to make the multipliers of «", y^, zi integer 
functions : for if we look to any argument in JJ, V, or W, it is of the form 
os^y^'z", and one of the letters a, h, c must be not less than its correspondent 
a, ^, y, for otherwise a + b + c would be not greater than a + /3 + 7 — 3, 
that is, n would be not greater than (?i + 1) — 3, or w — 2, which is absurd : 
if now any one, as a, be equal to or greater than a, it may be made to 
supply an integer part to the multiplier of x\ 

Here it may be asked what is to be done with such terms as Kafly^sfi, 
when two letters a, 6 are each not less than their correspondents a, /3 : the 
answer is, such terms may be made to enter under the multiplier of «", 
or of a^, or to supply a part to both in any proportion at pleasured. 

From the equations above we get, by linear elimination, 

FG'E" + GH'F" + HF'G" - GFH" - HG'F" - FH'G" = 0. 
This may be denoted thus : 11 (a, /3, 7) = 0, which equation I call a secondary 
derivative, and the left side of it a secondary derivee ; a, ^,j may likewise 
be termed the indices of derivation (as r, s, t, &c. are of augmentation). 

Now since a + /3 + 7 = w + 1, it is clear that the index of 11 (a, /3, 7) 
is always n + n +n — (n+1); that is, 2m — 1. 

* See for Latin translation tlie preceding note. 

t The prefixes of any such terms (say K) may be conceived as made up of two parts, an 
arbitrary constant, as e and (iT - c) ; e will disappear spontaneously from the final derivee. 



78 On a linear Method of Elimmating between [15 

1st. Let any two of the indices of derivation be taken zero, then it is 
easily seen that all the terms in 11 (a, yQ, ly) vanish, and consequently the 
secondary derivative equations obtained upon this hypothesis become mere 
identities, and are of no use. 

2nd. Let any one of them become zero. 

It is manifest, from the doctrine of simple equations, that 11 (a, /S, 7) may 
be made equal to 



X'U+fx'V+v 

\"U+ijl"V+v"w\~ , 
) x^ 

upon the understanding that 

\ = G'H" -Q"H', iJ. = H'F"-H"F', v = F'G"-F"G', 
X' = G"H - GH", fj,' = H"F - HF", v' = F"G- FG", 
X" = GH'-G'H, fj,"=HF'-H'F, v" = FG'-F'G. 
The three rows of coefficients will be respectively of the degrees 

(w-/3) + (?i-7), («-7) + (n-a), {n - a) + {n - ^). 
Thus if any one of the indices a, /3, 7 be zero, 11 (a, /S, 7) becomes 
identical with X'U + fJV + v'W, where the multipliers of U, V, W are of 
2n — (a + /3 + y) dimensions, that is of (w — 1) dimensions, and may accord- 
ingly be put under the form 

that is to say, becomes a linear' function of the augmentatives, and therefore 
if combined with them in the process of linear elimination would give rise 
to the identity = 0. 

Hence we must reject all such secondary derivatives as have zero for one 
of the indices of derivation. But all others, it may be shown, will be linearly 
independent of one another, and of the augmentees previously found. Hence, 

besides 3 — - equations of augment of the degree 2?i — 1, we shall have 

of the same degree so many equations of derivation as there are ways of 
stowing away between three lockers (?i+l) things, under the condition that 



no locker shall ever be left empty, that is 



2 



, . „ , n-1 ^n{n + l) 2w(2w + l) 
Thus, then, m all we have n — g— + 3 -^~^ — -' = — ^-^ equations, 

which is exactly equal to the number of arguments to be eliminated. Hence 

* Vide page 76 for the Latin version. 



15] double, treble, and other Systems of Algebraic Equations. 79 

the final derivee can be obtained by the usual explicit rule of permutation, 
and moreover will be its lowest form, for it will contain in each term 

^ — - prefixes belonging to the augmentatives of U, and a like number 

n—1 
pertaining to those of V and of W, as well as n — ^ — belonging to the 

secondary derivatives, each prefix in any one of which is triliteral, containing 
a prefix drawn out of those belonging to each of the proposees. 

Thus every member containing n —^ h n , that is n^ of the original 

prefixes belonging to U, V, W, singly and respectively, the final derivee 
evolved by this process will be in its lowest terms ; as was to be proved. 



Case A. Indices all equal. 
Method 2. 

It is remarkable that we may vary the method just given by making 

7- + r' + r" = n — 2, s + s' + s" = n — 2, t + t' + 1" = n - 2. 
The augmentatives will thus be of the degree 2n — 2. 

Furthermore, we must make a + /3 + 7 = ?i + 2. It will still be possible 
to satisfy by integer multipliers the equations 

U^i^F+y^F' + zyF", 

V=x''G + y^G' + ziG", 

W=x-^H + ym' + zyH'\ 

[these it will be useful in future to term the equations, x"-, y^, z^ being the 
arguments, and F, G, H, &c. the factors of decomposition] for otherwise 
calling the indices of x,y,ziD. any original argument a, h, c, their sum or n 
would be not greater than (w + 2) — 3, that is {n — 1), which is absurd. 

For the same reasons as in the last case no index of augmentation must 

be made zero : the degree of each will be (n — a) + (?i — /3) + {n — <y), that is 

(w + 1 ) w 
(2?i — 2), and their number „ ; the number of augmentatives will be 

— ^^ — - — — linearly uninvolved, each of the degree 2n — 2, and therefore 

. . (2n -l)2n ^ 

contammg -^^ ^— ^ — arguments. 

Now in+l)n ^ S (n-l)n ^( 2n -1) 2n 



80 On a linear Method of Eliminating between [15 

Hence the final derivee may be found, and it will be in its loivest terms, 
for every member will contain — ^ — - — — letters due to the augmentative, 

and -— ^ — ^ — ~ due to the partial derivative equations; in all then there will 

be 3m- letters in each term. 

This second method being applied to three quadratic equations of the 
most general form, leads to the problem of eliminating between six simple 
equations which lies within the limits of practical feasibility, and it is my 
intention to register the final derivee upon the pages of some one of our 
scientific Transactions as a standing monument for the guidance of hereafter 
coming explorers*. 



Scholium to Case A. 

If we attempt to carry forward these processes to quaternary systems, it 
becomes necessary to make 

a + /S + 7 + S = (r-2))i+l 

or else a + /3 + y + S = {r-2)n + 2, 

where r is the number of proposees. 

Now if the factors in the equations of decomposition are all integer, 
one of the indices of derivation must be not greater than the corresponding 
index in any of the original arguments, which may easily be shown to be 
always impossible for a system of equations, complete in all their terms, 
whenever their number r is greater than three, ifa + )3+'y+S = (i — 2)?i+2; 
but if a + ^+y+B = {r—2)n+l only possible for the case of m = 2. 



Particular method applicable to four Quadratics. 

Let 11=0, V=0, 1^=0, Z = 0, be four quadratic equations existing 
between x, y, z, t. 

Make xU=0, a:V=0, xW=0, xZ=0, 

yU=0, yV=0, yW=0, yZ=0, 

zU=0, zV=0, zW = 0, zZ=0, 

tU = 0, tV=0, tW=0, tz = o. 

* Elimination between two quadratics leads to a final derivee made up of seven terms only ; 
the final derivee of three quadratics is made up of at least several thousand ; nay, I believe I may 
safely say, several myriads of terms ! 



15] double, treble, and other Systems of Algebraic Equations. 81 

Also write U = x-F + yF' +ZF" + tF'" = 0, 

V=x-G+yG' + zG" + tG"' =Q, 

W = 'x'H + yH' + zH" + tH'" = 0, 

^ = «;=^ + yK' + zK" + tK'" = 0. 

By elimintating linearly we get 

2 {F^G' {H"K"' - H"'K")\ = 0, 

which will be of the third degree, since the factors represented by the 
unmarked letters F, G, H, K are of zero, and all the rest of unit dimensions. 

Similarly we may obtain other equations, so that besides the sixteen 
augmentatives already written down, we have four secondary derivatives, 
namely, 

n(2iii)=o, n(i2ii) = o, n(ii2i) = o, n(iii2)=o. 

Thus we have twenty equations and as many arguments to eliminate, since 
a perfect cubic function of four letters contains twenty terms. 

The final derivee will contain 16 + 4 . 4 letters, that is 32, 8 or 2^ belonging 
to each system of original prefixes in each member, and will therefore be in 
its lowest terms : for one of the canons of form teaches us, a priori, that 
every member of the derivee deduced from any number of assumed equations 
must contain in each member as many prefixes belonging to one equation of 
the system as there are units in the product of the indices of all the rest 
taken together. 

Corollary to Case A. 

Either of the two methods given as applicable to this case enables us to 
determine integer values of X, Y, Z, which shall satisfy the equation 
■ XU+YV+ZW^FxPyiz-, 

where F is the final derivee and p + q + r = 3n — 2. For by the doctrine of 
simple equations we know how to express F in terms of the linear functions, 
out of which it is obtained by permutation, that is we are able to assign 
values of A, B, C, and their antitypes, as also of L and its antitype, which 
shall satisfy the equation 

2 (Ax'-y'-'z^'U) + 2 {BxYz^'T) + 2 (Cx*y''z'"W) 

+ 2 {in (a, /3, 7)} = Fxfy^z'^, (1) 

where A, B, C, as well as L and all the quantities formed after them, are 
made up of integer combinations of the original prefixes. 

Now the functions 11 (a, /3, 7) may be expressed in three ways in terms of 
U, V, W, as has been already shown. 



82 Ori a linear Method of Eliminating between [15 

We may therefore suppose these functions to be divided into three 
groups, and make 



w 



, ^ 8U+ S'V+S"W ,-, 

+ ^ -ooy-^ ■ (^) 

And it is evident that the equations (1) and (2) lead immediately to the 
equation 

XU+ YV + ZW = Fio^+f yo+a z'+\ 

if we call a, h, c the greatest values attributed respectively to a, /3, 7. 

Now if we suppose the first method to be followed, 

f+g + h=1n-\. 

And it will always be possible to make a, b, c of what values we please 
subject to the condition of a + b + c = n — l; for 07ie at least of the indices 
of derivation in 11 (a, ^, y) must be not greater than its correspondent 
among a,b, c; otherwise a + ^8 + 7 would be not less than {a + b + c) + S; but 

a + /3 + 7 = ?i+l 

a+ b + c =n — l, 
which is absurd. 

Hence we can satisfy XU + YV+ ZW = Fx^ yi z^ , p, q, r being subject to 
the condition of p + q + r=Sn — 2, but otherwise arbitrary. 

Moreover, we can not do so if p + q+r be less than 3?i — 2, for that 
would require a + b + c to be less than n — 1. Now if two of the indices 
of derivation, as a and /3, be made equal to a 4- 1, 6 + 1 respectively, the 
third y=(n + l) — {a+b + 2) = {71 — I) — (a + b), and is therefore greater 
than c : so that a + /3 + 7 for this case becomes greater than a + b + c, and 
the method falls to the ground. 

In fact, I have discovered a theorem which lets me know this, d priori, 
a law which serves as a staff to guide my feet from falling into error in 
devising linear methods of solution, and the importance of which all candid 
judges who have studied the general theory of elimination cannot fail to 
recognize. To wit, if X^, X^, X^.-.Xn be n integer complete polynomial 
functions of n letters cci, x„ ... Xn, and severally of the degree 61, b^, 63 ... 6„; 
then it is always possible to satisfy the identity 

P,X, + P^X, + P,X, + ...+ PnX^ = Fx,-^x,-'-x,-' ... a.-„«». 



15] double, treble, and other Systems of Algebraic Equations. 83 

if a, + tto + 0(3 + . . . + a„ be equal to or greater than &i + &» + &3 +...+&« — ?i + 1, 
but otherwise not*. 

This again is founded immediately upon a simple proposition, of which 
I have obtained a very interesting and instructive demonstration, shortly to 
appear, and which may be enumerated thus : " The number of augmentees 
of the same degree that can be formed, linearly independent of one another, 
Old of any number of polynomial functions of as m,any variables, may be 
either equal to or less than the number of distinct arguments contained in such 
augmentees, but never greater. The latter will be the case when the index 
of the atigmentees diminished by unity is less than the sum of the indices of 
the original unaugmented polynomials each so diminished; the former, when, 
the aforesaid index is equal to or greater than the aforesaid sum." 

To return to the particular case of finding X, Y, Z to satisfy 

XU+YV-vZW = FxPyiz\ 

This has been already done according to the first method ; if we employ 
the second method of elimination we shall have 

f-\-g + h = '2.n-2. 

But, now since a+yS+7 = ?i + 2, we shall easily see by the same method 
as above, that the least value of a-\-b + c (where a, b, c denote respectively 
the greatest values of a, /3, 7, appearing in the denominator of the fractional 
forms used to express 11 (ct, ^, 7)}, will be one greater than before, or n ; so 
that f+g + h + a + b + c will still be equal to 3m — 2, as we might, a priori, 
by virtue of our rule, have been assured. 

Ternary Systems. 

Case B. Two of the indices equal ; the third less by a unit. 

Let U=Q, F=0, W=0, be the three given equations severally of the 
degree n, n, {n— 1). 

* Hence it is apparent, that in applying the method of multipliers, a curious and important 
distinction exists between the oases of there being two equations, and there being a greater 
number to eliminate from : for in the first case the element of arbitrariness needs never to appear; 
in the latter it cannot possibly be excluded from appearing in the multipliers. 

This will explain how it comes to pass that the method of the text may be employed to give 
various solutions of the XU + YV+ ZW^^Fx'i'y'iz^ ; thus not only can p, q and r be variously 
made up of (f + a), [g + b), (h + c), but also n (a, /3, 7) when two of the indices (a, |8 suppose) are 
each not greater than the assigned greatest values a, b may be made to figure indifferently either 
under the form 

\U+f>.V+vW ^, ^ .yU+!i'V+v'W 

or that of . 

x" or 

6—2 



84 On a linear Method of Eliminating between [15 

Make r + r' + r" = n — 2, s + s' + s" = n— 2, t-^ t' + t" = n — 1, 

by multiplying U into x^y'^'z^'", V into x^y^' z^\ W into x*y*'z'", we obtain 
augmentees each of the same, namely, the {2n — 2)th degree. 

The number of these is 

{n — l)n (n—l)n n{n+l) 

2r ^ 2 "'' 2 ■ 

Again, make a + /3 + j=n + l. 

It will still be possible, as before, to form equations of decomposition in 
which 00°-, y^, zf are the arguments, and affected with integer factors. For if 
we look to W even, all its arguments are of the form x'^y^z', where 
a + 6 + c = (?i— 1), and each of these cannot be less than its correspondent, 
for that would be to say that (m — 1) is not greater (n + 1) — 3, a fortiori, 
U and V can be decomposed in the manner described. Thus, then, we 
shall obtain as many secondary derivees as in the last case (Method 1), 

that is, — ^-jr (since a + ^ + r^ is still equal to {n + 1)}, as before. More- 
over, each of these will be of (n — a) + (w — /3) + (?i — 1 — 7), that is of 2n — 2 
dimensions. 

Altogether, therefore, we have 

{n — \)n {n — V)n m(?i + 1)| {n — \)n 



linear independent equations of the degree 2n — 2, and the number of 

(2w — 1) 2n 
arguments to eliminate is ^^ ^ — . Now these two numbers are equal. 

„, ,.„,,. . . „ ^^, ^ . {n — \)n (n—l)n 
Thus we obtain a nnal derivee contammg 01 (7 s coetfacients ^ — ~ — H „- - , 

an equal number 01 K s, but of It s — -—^ — - + - — ^ — ; now »i(«— 1), 

n (n — 1) and jf exactly express the number that ought to appear of each 
of these respectively: hence the final derivee is clear of irrelevant factors. 



Ternary Systems. 
Case C. Two of the indices equal; the third one greater by a unit. 

Here, calling n the highest index, the augmentees must each be made 
of the degree {2n — 3), their number will evidently be 

(n—2)(n — l) {7i—l)n (n—l)n 



15] double, treble, and other Systems of Algebraic Equations. 85 

making the sum of the indices of derivation now, as before, equal to (w + 1) ; 
it will be still possible to form integer equations of decomposition, which will 
give rise to augmentatives of the degree (n — a) + (w — 1) — /S + (n — 1) — 7, 
that is, of (2)1 — 3) dimensions. The total number of equations, what with 
augmentatives and secondary derivatives, will be 

\n-2){n-\) (n-Vyn (n-l)n ) n(n-l) _ 4!n^-4<n+2 _ (2n-2)(2n-l) 



2 ' 2 ' 2 J 2 2 2 

that is, is equal to the exact number of distinct arguments contained between 
them. 

Also the final derivative will contain in each member 

(n - 2) (91 - 1) w(w-l) 
2 "^ 2 ' 

that is, (n — l)(n— 1), letters belonging to the first equation, and 

(to— l)w n(n—l) 
2 ■•■ 2 ' 

that is, n(n — l) belonging to those of the second and of the third, and will 
therefore be in its lowest terms. 



Corollary to Cases B and C. 

It is not necessary, after all that has been already said, to do more than 
just point out that the processes applicable to these cases enable us to deter- 
mine X, Y, Z, which satisfy the equation 

XU+YV+ZW = FxfyH\ 

where f+g + h = '?>n — 'i for Case B, 

and f+g + h = 3?i — 4 for Case C. 



16. 



MEMOIR ON THE DIALYTIC METHOD OF ELIMINATION. 
PART I. 

[Philosophical Magazine, xxi. (1842), pp. 534 — 539*.] 

The author confines himself in this part to the treatment of two equations, 
the final and other derivees of which form the subject of investigation. 

The author was led to reconsider his former labours in this department 
of the general theory by finding certain results announced by M. Cauchy in 
L'Institut, March Number of the present year, which flow as obvious and 
immediate consequences from Mr Sylvester's own previously published 
principles and method. 

Let there be two equations in x, 

U = acc^ + 6a;"-i + ca;"-^ + ex^-^ + &c. = 0, 

F = a«"» + /Sa;'"-' + r^x^'^ + &c. = 0, 

and let ra = m + t, where i is zero or any positive value (as may be). 

Let any such quantities as x^U, afV, be termed augmentatives of U or V. 

To obtain the derivee of a degree s units lower than V, we must join s 
augmentatives of U with s + i of F. Then out of 2s + t equations 

x'>U=Q, x'U=0, x"-U=0, x'-^U = 0, 

x'V = 0, «iF=0, x"~V=0, a;'+«-iF=0, 

we may eliminate linearly 2s + t — 1 quantities. 

Now these equations contain no power of x higher than m + i + s — 1 ; 
accordingly, all powers of x, superior to m — s, may be eliminated, and the 
derivee of the degree (m — s) obtained in its prime form. 

Thus to obtain the final derivee (which is the derivee of the degree zero), 
we take m augmentatives of U with n of F, and eliminate (m + n — 1) 
quantities, namely, 

X, x^, x^, up to a;"'+"~\ 

* Reprinted from Froc. Eoxj. Irish Acad., Vol. ii. (1840—1844), p. 130. 



16] On the Dialytic Method of Elimination. 87 

This process, founded upon the dialytic principle, admits of a very simple 
modification. Let us begin with the case where t = 0, or m = n. Let the 
augmentatives of U be termed U^, U^, U^, U3, ... and of V, 7„, Fj, V^, F3, ..., 
the equations themselves being written 

11= ax^ + hx^-^ + cx"'-^ + &c. 
V = a'«» 4- h'x^-^ + c'x'^-^ + &c. 
It will readily be seen that 
a'U^ — aVo, 

(b'Uo-bV„)+{a'U,-aV,l 
(c' U„-cV,)+{b'U,-b F) + (a/ U,-a V,), &c. 

will be each linearly independent functions of x, x-, ... x'"^^, no higher power 
of X remaining. Whence it follows, that to obtain a derivee of the degree 
(m — s) in its prime form, we have only to employ the s of those which occur 
first in order, and amongst them eliminate a;"^\ x'^~^, ... a;™'"*^'. Thus, 
to obtain the final derivee, we must make use of n, that is, the entire number 
of them. 

Now, let us suppose that t, is not zero, but m = n — i. The equation F 
may be conceived to be of « instead of m dimensions, if we write it under 
the form 

Oa;™ + Oa;»-i + Oa;"-" + . . . + Ox"'+' +ax"' + /Sa;™-! + &c. = 0, 
and we are able to apply the same method as above ; but as the first i of the 
coefficients in the equation above written are zero, the first i of the quantities 

(a' Uo-a F„), (b' Ul -bV,) + (a' U,-aV,], &c. 
may be read simply 

-aV„ -6F,-ciF, -cF„-6F-aF„&c. 

and evidently their office can be supplied by the simple augmentatives 
themselves, 

F„=0, V, = 0, F, = 0... F_i = 0; 

and thus i letters, which otherwise would be irrelevant, fall out of the several 
derivees. 

The author then proceeds with remarks upon the general theory of simple 
equations, and shows how by virtue of that theory his method contains a 
solution of the identity 

Z,C7+ YrV=D,. 
where B^ is a derivee of the rth degree of U and F, and accordingly, X,. of 
the form 

X+ fix + vai- + . . . + ^«"*~''~\ 

and Yr of the form 

l + mx + ...Jf tx"--^-^, 



88 . 071 the Dialytic Method of Elimination. [16 

and accounts a priori for the fact of not more than (?i— r) simple equations 
being required for the determination of the (m + n — 2r) quantities X, /i, v, &c. 
I, in, n, &c., by exhibiting these latter as known linear functions of no more 
than {n — r) unknown quantities left to be determined. 

Upon this remarkable relation may be constructed a method well adapted 
for the expeditious computation of numerical values of the different derivees. 

He next, as a point of curiosity, exhibits the values of the secondary 
functions, 

a'Ufj — aVo, 

b'U,-br,+ aU,-aV„ 

c'Uo - cFo + b'U,-hV, + aU, - aV,, &c. 

under the form of symmetric functions of the roots of the equations U=0, 
V = 0, by aid of the theorems developed in the London and Edinburgh 
Philoscphical Magazine, December 1839*, and afterwards proceeds to a more 
close examination of the final derivee resulting from two equations each of 
the same (any given) degree. 

He conceives a number of cubic blocks each of which has two numbers, 
termed its characteristics, inscribed upon one of its faces, upon which the 
value of such a block (itself called an element) depends. 

For instance, the value of the element, whose characteristics are r, s, is the 
difference between two products: the one of the coefficient rth in order 
occurring in the polynomial U, by that which comes sth in order in V; the 
other product is that of the coefiScient sth in order of the polynomial U, by 
that ?'th in order of F; so that if the degree of each equation be n, there will 
be altogether ^n(n + 1) such elements. 

The blocks are formed into squares or flats {plafonds) of which the 

number is ^ or — — — , according as n is even or odd. The first of these contains 

n blanks in a side, the next {n — 2), the next {n — 4), till finally we reach a 
square of four blocks or of one, according as n is even or odd. These flats 
are laid upon one another so as to form a regularly ascending pyramid, of 
which the two diagonal planes are termed the planes of separation and 
symmetry respectively. The former divides the pyramid into two halves, 
such that no element on the one side of it is the same as that of any block 
in the other. The plane of symmetry, as the name denotes, divides the 
pyramid into two exactly similar parts ; it being a rule, that all elements 
lying in any given line of a square {plafond) parallel to the plane of separation 
are identical; moreover, the sum of the characteristics is the same, for all 
elements lying anywhere in a plane parallel to that of separation. 

[* p. 40 above. Ed.] 



16] 



On the Dialytic Method of Elimination. 



89 



All the terms in the final derivee are made up by multiplying n elements 
of the pile together, under the sole restriction, that no two or more terms of 
the said product shall lie in any one plane out of the two sets of planes 
perpendicular to the sides of the squares. The sign of any such product is 
determined by the places of either set of planes parallel to a side of the 
squares and to one another, in which the elements composing it may be 
conceived to lie. 

The author then enters into a disquisition relating to the number of terms 
which will appear in the final derivee, and concludes this first part with the 
statement of two general canons, each of which affords as many tests for 
determining whether a prepared combination of coefficients can enter into 
the final derivee of any number of equations as there are units in that 
number, but so connected as together only to afford double that number, less 
one, of independent conditions. 

The first of these canons refers simply to the number of letters drawn out 
of each of the given equations (supposed homogeneous) ; the second to what he 
proposes to call the weight of every term in the derivee in respect to each of 
the vanables which are to be eliminated. 

The author subjoins, for the purpose of conveying a more accurate 
conception of his Pyramid of derivation, examples of the mode in which it is 
constructed. 



When n=\ there is one 



When 71 = 2 there is one flat, viz. 



1, 2 



2, 3 2, 4 
2. 4 3. 4 



Let n = 3, there will be two 
flats: 



Let n = 4, there will still be two 
flats only : 



2, 3 2, 4 

2, 4 3, 4 



1, 2 1, 3 1, 4 
1, 3 1, 4 2, 4 
1. 4 2. 4 3. 4 



1, 2 


1, 3 


1, 4 


1, 5 


1, 3 


1, 4 


1, 5 


2, 5 


1, 4 


1, 5 


2, 5 


3, 5 


1, 5 


2, 5 


3, 5 


4, 5 



90 On the Dialytic Method of Elimination. 

Let fi = 5, there will be three flats : 



[16 



2, 3 


2, 4 


2, 5 


2, 4 


2, 5 


3, 5 


2, 5 


3, 5 


4, 5 



Let n = 6, there will be three flats : 



3, 4 3, 5 

3. 5 4. 5 



1, 2 


1, 3 


1, 4 


1, 5 


1, 6 


1, 3 


1, 4 


1, 5 


1, 6 


2, 6 


1, 4 


1, 5 


1, 6 


2, 6 


3, 6 


1, 5 


1, 6 


2, 6 


3, 6 


4, 6 


1, 6 


2, 6 


3, 6 


4, 6 


5, 6 



2, 3 2, 4 2, 5 2, 6 

2, 4 2, 5 2, 6 3, 6 

2, 5 2, 6 3, 6 4, 6 

2. 6 3, 6 4, 6 5. 6 



1, 2 


1, 3 


1. 4 


1, 5 


1, 6 


1, 7 


1, 3 


1> 4 


1, 5 


1, 6 


1, 7 


2, 7 


1. 4 


1, 5 


1, 6 


1, 7 


2, 7 


3, 7 


1, 5 


1, 6 


1. 7 


2, 7 


3, 7 


4, 7 

5, 7 


1, 6 


1, 7 


2, 7 


3, 7 


4, 7 


1, 7 


2, 7 


3, 7 


4, 7 


5, 7 


6, 7 



Thus the work of computation reduces itself merely to calculating 

ji + 1 
n — g — elements, or the n{n + 1) cross-products out of which they are con- 
stituted, and combining them factorially after that law of the pyramid, to 
which allusion has been already made. 



17. 



ELEMENTARY RESEARCHES IN THE ANALYSIS OF 
COMBINATORIAL AGGREGATION. 

[Philosophical Magazine, XXIV. (1844), pp. 285 — 296.] 

The ensuing inquiries will be found to relate to combination-systems, 
that is, to combinations viewed in an aggregative capacity, whose species 
being given, we shall have to discover rules for ranging or evolving them in 
classes amenable to certain prescribed conditions. The question of numerical 
amount will only appear incidentally, and never be made the primary object 
of investigation*. 

The number of things combined will be termed the modulus of the system 
to which they belong. The elements taken singly, or combined in twos, 
threes, &c., will be denominated accordingly the monadic, duadic. triadic 
elements, or simply the monads, duads, or triads of the system. 

Let us agree to denote by the word synthemef any aggregate of com- 
binations in which all the monads of a given system appear once, and 
once only. 

It is manifest that many such synthemes totally diverse in every term 
may be obtained for a given system to an}' modulus, and for any order of 
combination. 

Let us begin with considering the case of duad synthemes. Take the 
modulus 4 and call the elements a, b, c, d. 

{ah, cd), (ac, bd), (ad, cb) constitute three perfectly independent 
synthemes, and these three synthemes include between them all the daad 
elements, so that no more independent synthemes can be obtained from them. 

* The present theory may be considered as belonging to a part of mathematics which bears 
to the combinatorial analysis much the same relation as the geometry of position to that of 
measure, or the theory of numbers to computative arithmetic ; number, place, and combination 
(as it seems to the author of this paper) being the three intersecting but distinct spheres of 
thought to which all mathematical ideas admit of being referred. 

t From ffvv and Ti9-np.i. 



92 Elementary Researches in the Analysis of [17 

Again, let a, h, c, d, e,/be the monads; we can write down five independent 
synthemes, to wit, 

ah, cd, ef\ 

ad, c/, eb 
ac, de,fb 
af, bd, ce 

ae, df, be ' 

We can write no more than these witliout repeating duads which have 
ah'eady appeared*. 

We propose to ourselves this problem : — A system to any evenf modulus 
being given, to arrange the whole of its duadsX in the form of synthemes ; or in 
other words, to evolve a Total of duad synthemes to any given even modulus^. 

When the modulus is odd, as before remarked, the formation of a duad 
syntheme is of course impossible, for any number of duads must necessarily 
contain an even number of monadic elements ; but there is nothing to 
prevent us from forming in all cases what may be termed a bisyntheme or 
diplotheme, that is, an aggregate of combinations, where each element occurs 
twice and no more. 

For instance, if the elements be called after the letters of the alphabet, 

, fab, be, cd, de, ea\ , , . , ■ , , , , . , • 

we nave 7,77, the bisynthematic total to modulus o ; and in 

\ac, ce, eb, bd, da/ 

* Such an aggregate of synthemes may be therefore termed a Total. 

t The modulus must be even, as otherwise it is manifest no single syntheme can be formed. 
We shall before long extend the scope of our inquiry so as to take in the case of odd moduli. 

t Triadio systems will be treated of hereafter. 

§ It is scarcely necessary to advert here to the fact of the problem being in general indeter- 
minate and admitting of a great variety of solutions. 

When the modulus is four there is only one synthematic arrangement possible, and there is 
no indeterminateness of any kind ; from this we can infer, a priori, the reducibility of a biquad- 
ratic equation ; for using <j>, f, F to denote rational symmetrical forms of function, it follows that 
if{t>(a,b), <t>{c,d)}\ 
F if {cp (a, c), (j> (6, d)}> is itself a rational symmetric function of a, b, c, d. 
lf{4.(a,d), 0(6, c)}) 
Whence it follows that if a, b, c, d be the roots of a biquadratic equation, /{0 (a, b), tp (c, d)} can 
be found by the solution of a cubic: for instance, {a + b)x (c + rf) can be thus determined, whence 
immediately the sum of any two of the roots comes out from a quadratic equation. 

To the modulus 6 there are fifteen different synthemes capable of being constructed ; at first 
sight it might be supposed that these could be classed in natural families of three or of five each, 
on which supposition the equation of the sixth degree could be depressed ; but on inquiry this 
hope will prove to be futile, not but what natural affinities do exist between the totals ; but in 
order to separate them into families each will have to be taken twice over, or in other words, 
the fifteen synthemes to modulus 6 being reduplicated subdivide into six natural families of five 
each. Again, it is true that the triads to modulus 6 (just like the duads to modulus i) admit of 
being thrown into but one synthematic total, but then this will contain ten synthemes, a number 
greater than the modulus itself. 



17] Combinatorial Aggregation. 93 

like manner 

ah, he, cd, de, ef, fg, go) 



ac, ce, eg, gh, hd, dj, fa^ the total to modulus 7. 

ad, dg, gc, cf, fh, he, 

92—1 

In general, if ?i be the modulus, the number of duads is n —^ — ; n being 

even, - duads go to each syntheme, and therefore the total contains (n — 1) 

of these. If n be odd, then, since always n duads go to a bisyntheme, the 

?i — 1 
number of such in the total is — ^— . 

Before proceeding to the solution of the problem first proposed, let us 
investigate the theory of diplothematic arrangement. Here we shall find 
another term convenient to employ. By a cyclotheme, I designate a fixed 
arrangement of the elements in one or more circles, in which, although for 
typographical purposes they are written out in a straight line, the last term 
is to be viewed as contiguous and antecedent to the first ; the recurrence 
may be denoted by laying a dot upon the two opened ends of the circle ; 
a.h.c .d . e will thus denote a cyclotheme to modulus 5; d.b .c .d. e.f.g.h.h 
the same to modulus 9; so also is a.h.c, d.e.f, g.h.k a cyclotheme of 
another species to the same modulus. In general the number of terms will 
be alike in each division of a cyclotheme. 

Now it is evident that every cyclotheme, on taking together the elements 
that lie in conjunction, may be developed into a diplotheme. Thus 

1 . 2 . S = 12, 23, 31, 

1 .2.3.4 = 12, 23, 34, 41, 

/1 2, 23, 31 
(1 . 2 . .3 ; 4.0.6; t . S . 9) = I 45, 56, 64 

\78, 89, 97 

Hence we shall derive a rule for throwing the duads of any system into 
bisynthemes. 

Let m = 3, we have simply dhc, 

in = 5, we write a . b . c . d . e, 

d . c .e.b .d, 

the second being derived from the first by omitting every alternate term ; 
similarly below, the lines are derived each from its antecedent. 

m = 7, wehave a .h.c .d .e.f.g, 

a .c. e. g .h .d.f, 
a . e .h.f.c.g .d. 



94 Elementary Researches in the Analysis of [17 

A very little consideration will serve to prove that in this way, m being 

a prime number, — ^ — cyclothemes may be formed, such that no element 

will ever be found more than once in contact on either side with any other ; 
whence the rule for obtaining the diplothematic total to any prime-number 
modulus is apparent. 

For example, to modulus 7 the total reads thus : — 
1st. ab, be, cd, de, ef, fg, ga\ 
2nd. ac, ce, eg, gb, bd, df, fa I , 
3rd. ae, eb, bf, fc, eg, gd, daj 
and no more remains to be said on this special case. 

Let us now return to the theory of even moduli, and show how to apply 
what has been just done to constructing a synthematic total to a modulus 
which is the double of a prime number. 

Suppose the modulus to be six, the number of synthemes is five. Let the 
six elements, a, b, c, d, e, f, be taken in three parts, so that each part contains 
two of them ; let these parts be called A, B, C, where A denotes ab, B, cd, 
and (7, ef. 

Now the duads will evidently admit of a distinction into two classes, 
those that lie in one part, and those that lie between two ; thus ab, cd, ef 
will be each unipartite duads, the rest will be bijMrtite. 

The unipartite duads may be conveniently formed into a syntheme by 
themselves; it only remains to form the four remaining bipartite duad 
synthemes. 

Write the parts in cyclothematic order, as below : 
ABC. 
It will be observed that each part may be written in two positions ; thus 

A may be expressed by , or by , 











B 


" 


" 


c 
d 


d 
" c 










C 


" 


" 


e 
f 


f 
" e' 


Now 


we 


may 


form 


a cyclic 


table 


of positions as 


below 














ABC 








1 1 1 
















122 


















212 


















2 21 







17] Combinatorial Aggregation. 95 

Here the numbers in each horizontal line denote the synchronic positions 
of the parts. 

On inspection it will be discovered that A will be found in each of its 
two positions, with B in each of its two ; similarly B with C, and G with A. 
In fact the four permutations, 11, 12, 21, 22, occur, though in different 
orders, in any two assigned vertical columns. 

Now develope the preceding table, and we have 

dee adf bcf bde, 

hdf bee ade acf; 

and these being read off (the superior of each antecedent with the inferior 
of each consequent*) must manifestly give the four independent bipartite 
synthemes which we were in quest of, videlicet 

(ad, cf, eh), {ac, de, fb), (bd, ce, fa), (be, df ea) ; 
these four, together with the syntheme first described (ab, cd, ef), constitute 
a duad synthematic total to modulus 6. 

Before proceeding further let ns take occasion to remark that the fore- 
going table of positions may evidently be extended to any odd number 
of terms by repetition of the second and third places, as seen in the annexed 
tables of position. 



i . 


.1. 


1.1 


.1 


1, 


.1. 


,1. 


.1, 


,1, 


,1. 


i, 


i, 


,2. 


.2.2, 


2 


1, 


,2, 


,2, 


.2, 


,2 


.2, 


.2t> 


2 


.1, 


.2.1, 


.2 


2 , 


.1, 


.2, 


.1 


.2 


.1 


.2, 


2 


.2 


.1.2 


.1 


2 


.2, 


.1 


.2, 


.1, 


.2, 


, i. 



Arrange in cyclothemes l'-^ — in number] the odd modulus system 



Now let 10 be the modulus. 

As before divide the elements into five parts, which call A, B, G, D, E. 

The unipartite duads fall into a single syntheme ; the eight remaining 
bipartite synthemes may be found as follows :— 

— — in nur . , 

A,B,C, D, E. We have thus 

ABGDE, 
AGEBD. 

* Any other fxed order of successive conjunction would answer equally well. 

t It will not fail to be borne in mind that in operating with these tables only contiguous 
elements are taken in conjunction : the first with the second, the second with the third, the 
third with the fourth, &c., and the last with the first ; no two terms but such as lie together are 
in any manner conjugated with one another. 



96 Elementary Researches in the Analysis of [17 

Let each cyclotheme be taken in the four positions given in the table 
above, we have thus 2x4, that is, eight arguments. 

db cde . a, 0y Be . abyde . a/S cBe, 

a^ry Se, oibcd e, a jSche, abyde, 

dcebd.dye/SB.dcebS.aye^d, 

ay e 0h, acebd, aye^d, ace b B. 

And each of these arguments will furnish one bipartite syntheme, by reading 
off, as before, the superior of each antecedent with the inferior of each 
consequent ; and the least reflection will serve to show that the same duad 
can never appear in two distinct arguments. 

In like manner, if the modulus be 14 and seven parts be taken, the 
bipartite synthemes, twelve in number, may be expressed symbolically thus : 

/ i.1.1.1.] .i.i\ 

. f A.B.C.D.E.F.G] 

+ 12 2 2 2.2.2 

+ 2.1.2.1.2.1.2(1 -I 

Ua.e.b.f.c.g.d} 

+ 2.2.1.2.1.2.1) 

A B 

Nay more, from the above table, if we agree to name the elements / n ^ "^c., 

we can at once proceed to calculate each of the twelve synthemes in question 
by an easy algorithm. For instance, 

(i.2.2.2.2.2.2)x(i.C.^.C?.5.i).F). 

= {A,C,, G,E„ E„_G„ G,B„ B„_D„ D,F„ F.A,). 



And again 



(2. 1. 2. 1.2. 1. 2) X (A. E.B.F.C.G.D) 
= A^E.„ E,B„ B„F„ FA, C^G„ G,D„ D,A,; 



each figure occurring once unchanged as an antecedent and once changed 
as a consequent. 

If it were thought worth while it would not be difficult, by using numbers 
instead of letters, to obtain a general analytical formula, from which all 
similarly constituted synthemes to any modulus might be evolved. 

But the rule of proceeding must be now sufficiently obvious ; the modulus 
being 2p, we divide the elements into p classes ; these may be arranged into 

-t- — - distinct forms of cyclothematic arrangement, and each of the cyclo- 

p — 1 

themes taken in four positions, thus giving 4 X'^ — , that is, 2p—2 bipartite 

synthemes, the whole number that can be formed to the given modulus 2p. 



17] Combinatorial Aggregation. 97 

I shall now proceed to the theory of bipartite synthemes to the modulus 
2m X p, by which it is to be understood that we have p parts each containing 
2»n. terms, and p is at present supposed to be a prime number; the total 
number of synthemes to the modulus 2mp being %np — 1, and 2m — 1 of 
these evidently being capable of being made unipartite ; the remainder, 
2mp — 2m, that is, {p — 1) 2m, will be the number of bipartites to be 
obtained*: 

2m (jo - 1) = -^-n— X 4m ; 

»— 1 

^—-T — denotes the total number of cyclothemes to modulus p ; 4m, as will be 

presently shown, the number of lines or syzygies in the Table of position. 
To fix our ideas let the modulus be 4 x 3, and let A, B, G be three parts : 

Oi a^ cts di] 

bi &2 ^3 ^4 \ their constituents respectively. 

Ci C2 C3 C4} 

Give a fixed order to the constituents of each part, then each of them may 
be taken in four positions ; thus A may be written 

Oiaia^tti, 
asaiOia^, 

Assume some particular position for each, as, for instance, 

^^2 "2 Cg , 

aibiCi, 

and read off by coupling the first and third vertical places of each ante- 
cedent with the second and fourth respectively of each consequent ; we have 
accordingly, 

Oi&j, 61C2, Cia^, 

a^bi, &3C4, Cstti. 

It is apparent that the same combinations will recur if any two contiguous 
parts revolve simultaneously through two steps ; or in other words, that 
A^Bs — ^r+2-Ss+2) where /j, is any number, odd or even. 

* In general, if there be ir parts of fi. terms each, and fiir be even, the number of bipartite 
synthemes is {ir - 1) yn, as is easily shown from dividing the whole number of bipartite duads 
by the semi-modulus. 

S. 7 



or more generally, 
So that 



98 Elementary Researches in the Analysis of [17 

Symbolically speaking, therefore, as regards our table of position, 
r:s = r + 2:s + 2, 

= r + 2 + 4i : s + 2 + 4i. 

1:1 = 3:3, 2:1 = 4:3, 

1:2 = 3:4, 2.2 = 4:4, 

1:3 = 3.1, 2.3 = 4:1, 

1:4 = 3.2, 2.4 = 4:2. 

There are therefore no more than eight independent unequivalent permuta- 
tions to every pair of parts. Now inspect the following table of position : — 

i . 1 . 1, 2.1.2, 

1.2.3, 2.2.4, 
1.3.2, 2.3.1, 

1.4.4, 2.4.3. 

It will be seen that in the first and second, second and third, third and 
first places, all the eight independent permutations occur under different 
names ; the law of formation of such and similar tables will be explained 
in due time ; enough for our present object to see how, by means of this 
table, we are able to obtain the bipartite synthemes to the given modulus 

4x3; the number according to our formula is 2 x 4 x — — = 8, and they 
may be denoted symbolically as follows : — 

: ^ /I. 1.1 + 1. 2. 3 + 1. 3. 2+1.4. 4\ 

^^+ 2. 1.2 + 2. 2. 4 + 2. 3. 1 + 2. 4. 3/ ■ 

Each of the eight terms connected by the sign of + gives a distinct syn theme; 
for example, let us operate on 

i . 5 . X (2 . 3 . 1). 

2.3.1 denotes 2.3, 3.1, 1.2. 

2 . 3 gives rise to 2 (3 + 1) + (2 + 2) . (3 + 3) = 2 . 4 + 4 . 2. 
8 . 1 gives rise to 3 (1 + 1) + (3 + 2) . (1 + 3) = 3 . 2 + 1 . 4, 
1 . 2 gives rise to 1 (2 + 1) + (1 + 2) . (2 + 3) = 1 . 3 + 3 . 1. 
The syntheme in question is therefore 

A,B,, A,B„ B,a, B,C„ C,A„ G,Ai, 
and so on for all the rest, the rule being that 

r : s = r (s + 1) + (r + 2) (s + 3). 



r A.B.G.D.E.F.G 
\+ A.C.E.G.B.D.F 
\+ A.E.B.F.C.G.i) 



< 



17] Combinatorial Aggregation. 99 

Now, as before, it is evident that if we look only to contiguous terms, the 
above table of position may be extended to any number of odd terms, simply 
by repetition of the second and third figures in each syzygy ; and hence the 
rule for obtaining the bipartite synthemes to the modulus 4 x jp is apparent. 

7-1 

For instance, let p = T, there will be 8 x — ^ , that is, 8 x 3 of them denoted 

as follows : — 

1. 1. 1. 1.1. 1.1 + 2. 1.2.1. 2. 1.2^ 
1+1.2.3.2.3.2.3 + 2.2.4.2.4.2.4' 
+ 1.3.2.3.2.3.2 + 2.3.1.3.1.3.11'" 
1+1.4.4.4.4.4.4 + 2.4.3.4.3.4.3) 
As an example of the mode of development, let us take the term 
i.^.5.^.0.G'.I)x2.4.3.4.3.4.3, 
2.4.3.4.3.4.3 = (2:4, 4:3, 3:4, 4:3, 3:4, 4:3, 3:2) 

, 2'.1) 4.4] 3.1) 4.4) 3.1) 4.41 3.3) 
U 4 . 3} + 2 . 2J + 1 . 3} + 2 . 2J + 1 . 3J + 2 . 2j + 1 . Ij 

A.E.B.F.G.G.b = A.E, E.B, B.F, F.C, G.G, G.D, D.A, 
and the product 

^fA,E„ E,B„ B,F„ F,C„ C,G^, G,D„ I),A^\ 
[a,e\, E,B„ B,F„ F.G„ C,G„ G,D,, d,aJ' 

Let the modulus be 6x3; as before, give a fixed cyclic order to the 
constituents of each part, and each will admit of being exhibited in six 
positions. 

Write similarly as before, 

«i ^1 Ci , 

(^3 ^3 ^3 ) 

a-B h Cb, 

as bs Ce, 

and take the odd places of each antecedent with the even places of each 
consequent ; it will now be seen that 

7-:s = r + 2:s + 2 = r+4:s + 4, 

and the number of independent permutations is —^ =2.6; and so in 
general, if there be 2m constituents in a part, the number of independent 

, , . . 2m . 2ni ^ 
permutations is = 4m. 

-^ 771 

2 

7—2 



LofC. 



100 



Elementary Researches in the Analysis of 



[17 



The rule for the formation of the table will be apparent on inspection. 
I suppose only three parts, as the rule may always be extended to any 
number by reiteration of the second and third terms. The table will be 
found to resolve itself naturally into four parts, each containing m lines. 



Let m= 1, we have 








1.1.1 


2.1.2 




1.2.2 


2.2.1 


m = 2, we have 








1.1.1 


2.1.2 




1.2.3 


2.2.4 




1.3.2 


2.3.1 




1.4.4 


2.4.3 


m = 3, we have 








1.1.1 


2.1.2 




1.2.3 


2.2.4 




1.3.5 


2.3.6 




1.4.2 


2.4.1 




1 . .5 . 4 


2.5.3 




1.6.6 


2.6.5 


m = 4, we have 








1.1.1 


2.1.2 




1.2.3 


2.2.4 




1.3.5 


2.3.6 




1.4.7 


2.4.8 




1.5.2 


2.5.1 




1.6.4 


2.6.3 




1.7.6 


2.7.5 




1.8.8 


2.8.7 



So that x, going through all its values from 1 to m, the general expression 
for the four parts is 

l.x(2x-l) + l{7n+x)2x ]■ 

+ 2 . « . 2« + 2 (m + a-) (2a; - 1)\ ' 

To show the use of this formula, let us suppose that we have seven parts, 
each containing ten terms, the general expression for the bipartite duad 
synthemes is 

/ l.x(2x-l)x(2x-l)x(-2x-l) \ 

A.B.C.D.E.F.G] ^0900 

+ 1 .X . Ax .X .2x .X. 2x • I 

+ A.E.B.F.C.G.D] +-2{o + x){2x-l)ib + x)(2x-l){b+x){2x-\)] 



17] 



Gomhinatorial Aggregation. 



101 



Make, for example, a; = 3, one of the synthemes in question out of the 
twelve corresponding to this value will be 

A.G.E.G.B.D.Fx'2..2,.Q.Z.Q.2,.Q. 
Here 

A.G.E.G.B.D.F=AG, CE, EG, GB, BD, DF, FA, 

2.3.6.3.6.3.6 = 



= 2.4 3.7 


6.4 


3.7^ 


6.4 


3 


7 


6 


3 


+ 4.6 +5.9 


+ 8.6 


+ 5.9 + 8.6 


■ + 5 


9 


+ 8 


5 


+ 6.8 (- + 7.1 


+ 10.8 


^- +7.1 >- +10.8 


- + 7 


1 


r + 10 


7 


+ S.IOJ +9.3 


+ 2.10 


+ 9.3 


+ 2.10 


+ 9 


3 


+ 2 


9 


+ 10.2 ' +1 .5 


+ 4.2 


+ 1.5 


+ 4.2 


+ 1 


5, 


+ 4 


1 


and the product 










= A,C„ G,E„ E,G„ G,B„ B,D„ I),F„ 


F,A, 








AiGs, GsEg, EgGe, GsBg, B^D^, D^F^, 


FsA, 










&c. 


&c. 




&c. 









To prove the rule for the table of formation, it will be sufficient to show 
that no two contiguous duads ever contain the same or equivalent permutations ; 
the equation of equivalence it will be remembered is 

r:s = r + 2i± 2m :s + 2i± 2m. 

Now, as regards the first and second terms, it is manifest that 1 : x cannot be 
equivalent, either to 1 :x' nor to 2 : x, nor to 2 : x', where x' is any number 
differing from x. 

Similarly, as regards the last and first terms, x : 1 cannot be equivalent to 
x' : 1, nor to « : 2, nor to x' :2; therefore there ,is no danger as far as the first 
term is concerned, either as antecedent or consequent. 

Again, it is clear that x : (2x — 1) cannot interfere with x : 2a?', nor 
(m + x): 2x with (m + x') : (2a;' — 1) ; neither can (2a; — 1) : a; with 2a;' : x', nor 
2x:(ni + x) with (2a;' — 1) : (m + x'). 

Again, if possible, let 

X : {2x - 1) = (m + x') : (2a;' - 1); 



then 

and 

therefore 



in + x' — x = 2i, 
2x' — 2x= 2i, 
2in = 2i, 



which is impossible, since + i is the difference between two indices, each 
less than m. 



102 Researches in Analysis of Combinatorial Aggregation. [17 

Similarly, 

?ft + a; : 2a; cannot = x' : 2x', 

and vice versa with the terms changed 

2x : (m + x) cannot = 2a;' : «', 
and 

(2a; — l):x cannot = (2a;' — 1) : (m + x), 

which proves the rule for the table of formation. 

So much for the bipartite duad synthemes. As regards the unipartite 
synthemes little need be said, for every part may be treated as a separate 
system, and as each will produce an equal number of synthemes, these being 
taken one with another, will furnish just as many unipartite synthemes of 
the whole system as there are synthemes due to each part. Thus then the 
synthematic resolution of the modulus 2m x p may be made to depend on 
the synthematization of 2m and the cyclothematization of p. This has been 
already shown (whatever m may be) for the case of p being a prime number ; 
but I proceed now to extend the rule to the more general case of p being 
any number whatever. 



18. 



ON THE EXISTENCE OF ABSOLUTE CRITERIA FOR DETER- 
MINING THE ROOTS OF NUMERICAL EQUATIONS. 

[Philosophical Magazine, XXV. (1844), pp. 442 — 445.] 

I WISH to indicate in this brief notice a fact which I believe has escaped 
observation hitherto, that there exist, certainly in some cases, and probably 
in all, infallible criteria for determining whether a given equation has all its 
roots rational or not. 

In the equation of the second degree it is enough, in order that this may 
be the case, that the expression for the square of the difference of the roots 
shall be a perfect square ; in other words, if x^ — px + q=0 have its roots 
rational, p" — 4g must be not only a positive number (the condition of the 
roots being real), but that number must also be a complete square. In this 
case it is further evident that p must be either prime to q, or if not, the 
greatest common measure of p^ and q must be a perfect square ; but this 
condition is contained in the former, which is a sufficient criterion in itself 

If we now consider the equation of the third degree, 

x^ —poo- + qx — r = 0, 

one condition is, that the product of the squared differences shall be a perfect 
square ; in other words, the equation cannot have all its roots rational 
unless 

p2^2 _ 4^3 _ iSpqr — 4<p^r — 27r- 

be a positive square number. 

This remark is made at the end of the second supplement of Legendre's 
Theory of Numbers, and is indeed self-evident ; and in like manner one 
condition may be obtained for an ecfuation of any degree which is to have 
all its roots rational ; but this is far from being the sole condition required. 



104 On Absolute Criteria for determining [18 

In the equation of the third degree, however, one other condition, conjoined 
with that above expressed, will serve to determine infallibly whether all the 
roots are rational or not. 

To obtain this condition, let us suppose that by making dx = y+p we 
obtain the equation 

y^-Qx-R = Q. 

Calling the three roots of this new equation a, ^, 7 (all of which it 
is evident must be rational if those of the first equation are so), we have 

a + /3 + 7 = 0, 

Q = - (a/3 + a7 + (87) = a^ + a/3 + /3-, 

R= a/37. 

From the last two equations it is easily seen that if k be any prime factor 
common to Q and -K, k"^ will be contained in Q, and k^ in R; or, in other 
words, k will be a common measure of a, /3, 7. 

We have therefore a second condition, that dq — ^p' shall be a negative 
quantity, which is either prime to 2p^ — 9qp + 27r, or else so related to it, 
that the greatest common measure of the cube of the first and the square 
of the second is a perfect sixth power. 

I now proceed to show the converse, that if these two conditions be both 
satisfied (and it will appear in the course of the inquiry that the first does 
not involve the second), the roots cannot help being all rational. 

It is evident that the two conditions in question are tantamount to 
supposing that the roots of the proposed equation are linearly connected 
with those of another z^ — Qz—R = (by virtue of the assumption 3a; = kz +p), 
where Q may be considered as prime to R ; and where 4Q^ — 27i?^ is a 
perfect square. 

Let now 4Q= - 21 R- = D\ then D^ + 21 R- = 4Q^ or D^ + 3 {^Rf = 4Ql 

Here, as Q is prime to R, D can have no common measure but 3, 
with 3E. 

Firstly, let Q be prime to oR. 

Then putting /^+ 3^"= Q', the complete solution of the equation im- 
mediately preceding is contained in the two systems : 

1st. D=2f, SR = 2g. 

2nd. D = (f±3g), SR=f+g, 

and for both systems, 

f±9>^(-S) = {h±Sk^(-3)}\ 



2g-| the Roots of Numerical Equations. 105 

The s scond system must thei-efore be rejected, for g evidently contains 3, 
and there '^°'^® /~ ^^ - 3 ^^'■^ contain 3, and therefore D and therefore Q will 
do the san ^®' contrary to supposition. 



Henc "^ 



v'SVfd-?)!] 
=\7{i±?v'(4)} 
-^7{iV(4)} 

^{/±5'V(-3)} 



"3V(-3) 

= -^±3VF3) = ^±^^^-'^= 
and the three roots of the equation being 
{\ + /.V(-3)}+{X-/.V(-3)}, 

— 2 1^ + /* V(- 3)1 + 2 y^~l^ V (- 3)), 

will evidently be all rational, which of course includes the necessity of their 
being also integer. 

Again, secondly, if we suppose that Q does contain 3, D'^ will contain 27, 
and consequently D will contain 9 ; and we shall have 

Here R being prime to tt , it may be shown, as in the last case, that the 
complete solution is 

f ±^V(-3)=(/.±i.v/(-3)h 
consequently 

and the three roots of the equation are 

2A, h - 3k, h + 3k 

respectively, and are therefore all rational. 

Here it may be observed that the condition of R being an even number, 
which we know, d priori, is the case when all the roots are rational, is 



106 On Absolute Criteria of Numerical Equatiok "- 

involved in the two more general conditions already expressed. '' . . 

be evident that the first condition by no means involves the seer ' 
perfectly easy to satisfy the equation f- + Zg- = (^^ without supposiP ^^^ ^^^^ 
relative to h, the common measure of/, g, Q, except that it be iti 
form X^ + Syti'^, which will give 

an equation which can be solved in rational terms for all values of X, n, r, I 
and consequently the product of the squares of the differences of the roots, 
may be a square, and at the same time the roots themselves may be 
irrational*. 

I believe it will be found on inquiry that the equation x'^ — qx + r = 
will always have two rational roots if 

be a complete square, provided that q be prime to r. . 

Furthermore, viewing the striking analogy of the general nature of the 
conditions of rationality already obtained, to those which serve to determine 
the reality of the roots of equations, I am strongly of opinion that a theorem 
remains to be discovered, which will enable us to pronounce on the existence 
of integer, as Sturm's theorem on that of possible roots of a complete equation 
of any degree : the analogy of the two cases fails however in this respect, 
that while imaginary roots enter an equation in pairs, irrational roots are 
limited to entering in groups, each containing two or MOHE. 

* Thus then it appears that the total rationality of the roots of the equation x^-qx-r=0 
may be determined by a direct method without having recourse to the method of divisors to 
determine the roots themselves ; the two conditions being that iq^—27r^ shall be a perfect square, 
and the greatest common measure of q'^ and r- a perfect sixth power. 



19. 



AN ACCOUNT OF A DISCOVERY IN THE THEORY OF 
NUMBERS RELATIVE TO THE EQUATION Aaf + Bif+Gz'= Dxyz. 

{Philosophical Magazine, XXXI. (1847), pp. 189 — 191.] 

First General Theorem of Transformation. 

If in the equation 

Ax' + By^+ Cz' = Dxyz, (1 ) 

^•1 and B are equal, or in the ratio of two cube numbers to one another, and 
if 27 ABC — D^ (which I shall call the Determinant) is free from all single or 
square prime positive factors of the form 6n + 1, but without exclusion of 
cubic factors of such form, and if A and B are each odd, and C the double or 
quadruple of an odd number, or if A and B are each even and G odd, then, 
I sa}^, the given equation may be made to depend upon another of the form 

A'u^ + B'v' + G'w'^ = D'uvw ; 
where 

A'B'G' = ABC, 

D' = D, 

uviu = some factor of z. 

The following are some of the consequences which I deduce from the 
above theorem. In stating them it will be convenient to use the term Pu.re 
Factorial to designate any number into the composition of which no single or 
square prime positive factor of the form 6w,+ l enters. 

The equations 

a? + 2/'' + 2z^ = Dxyz, 

a? + y^ + 4^^ = Dxyz, 

2a.'' + 22/3 + z^ = Dxyz, 

are insoluble in integer numbers, provided that the Determinant in each case 
is a Pure Factorial. 



108 On a Discovery in the Theory of Numbers. [19 

The equation 

«=> + 2/' + -42^ = %Bxyz 

is insoluble in integer numbers, provided that the Determinant, for which in 
this case we may substitute A — 275^ is a pure factorial whenever A is of 
the form 9n + 1, and equal to 2/;^'*^ or 4^^'*', j) being any prime number 
whatever. 

I wish however to limit my assertion as to the insolubility of the 
equations above given. The theorem from which this conclusion is 
deduced does not preclude the possibility of two of the three quantities 
X, y, z being taken positive or negative units, either in the given equation 
itself or in one or the other of those into which it may admit of being 
transformed. Should such values of two of the variables afford a particular 
solution, then instead of affirming that the equations are insoluble, I should 
affirm that the general solution can be obtained by equations in finite 
differences*. 

Second General Theorem of Transformation. 

The equation 

f^af + g^y^ + h^z^ = Kxyz (2) 

may always be made to depend upon an equation of the form 

Au^ + Btfl + Giifi = Duvw, 
where 

ABC = R'-S\ 

and iivw = some factor of fa; + gy + hz. 

R representing K + Qfgh, 
S „ K-3fgh. 

* Take for instance the equation x^ + y^ + 2z^=dxyz. The Determinant 27.25 is a Pure 
Fa,ctorial : consequently if the solution be possible, since in this case the transformed must 
be identical with the given equation, this latter must be capable of being satisfied by making x 
and y positive or negative units. Upon trial we find that .■r = l, i/ = l, z = 2 will satisfy the 
equation. I believe, but have not fully gone through the work of verification, that these are the 
only possible values (prime to one another) which will satisfy the equation. Should they not be 
so, my method will infallibly enable me to discover and to give the law for the formation of all 
the others. 

Here, then, under any circumstances, is an example, the first on record, of the complete 
resolution of a numerical equation of the third degree between three variables. 



19] On a Discovery in the Theory of Numbers. 109 

I have not leisure to show the consequences of this theorem of trans- 
formation in connexion with the one first given, but shall content myself 
with a single numerical example of its applications : 

n" + y" + z" = — Qxyz 
may be made to depend on the equation 

w' + v^ + 'uf = 0, 
and is therefore insoluble. 

It is moreover apparent that the Determinant of equation (2) trans- 
formed is in general — 27i?'', and is therefore always a Pure Factorial, and 
consequently the equation 

f'^0!? + g^y" + h?z^ = Kxyz 

will be itself insoluble, being convertible into an insoluble form, provided that 
K-\-Qfgh is divisible by 9, and provided further that {K+Qfghy — {K — Sfghy 
belongs to the form nv'Q, where Q is of the form 9n ±1, and also of one or the 
other of the two forms 2p^^'^^, 4p^''^\ p being any prime number whatever. 

Pressing avocations prevent me from entering into further developments 
or simplifications at this present time. 

It remains for me to state my reasons for putting forward these dis- 
coveries in so imperfect a shape. They occurred to me in the course of 
a rapid tour on the continent, and the results were communicated by me to 
my illustrious friend M. Sturm in Paris, who kindly undertook to. make them 
known on my part to the Institute. 

Unfortunately, in the heat of invention I got confused about the law of 
oddness and evenness, to which the coefficients of the given equation are in 
the first theorem generally (in order for the successful application of my 
method as far as it is yet developed) required to be subject. I stated this 
law erroneously, and consequently drew erroneous conclusions from my 
Theorems of Transformation, which I am very anxious to seize the earliest 
opportunity of correcting. I venture to flatter myself that as opening out 
a new field in connexion with Fermat's renowned Last Theorem, and as 
breaking ground in the solution of equations of the third degree, these 
results will be generally allowed to constitute an important and substantial 
accession to our knowledge of the Theory of Numbers. 



20. 



ON THE EQUATION IN NUMBERS Ax' + Bif + Cz^ = Dxyz, AND 
ITS ASSOCIATE SYSTEM OF EQUATIONS. 

[Philosophical Magazine, XXXI. (1S47), pp. 293 — 296.] 

In the last Number of this Magazine I gave an account of a remarkable 
transformation to which the equation 

Ax^ + Bif + Gz^ = Dxyz 

is subject when certain conditions between the coefficients A, B, C, D ai-e 
satisfied ; which conditions I shall begin by expressing with more generality 
and precision than I was enabled to do in my former communication. 

1. Two of the quantities A, B, G are to be to one another in the ratio of 
two cubes. 

2. 27 ABG — D^ must contain no positive prime factor whatever of the 
form 6n + l. I erred in my former communication in not excluding cubic 
factors of this form. 

3. If 2™ is the highest power of 2 which enters into ABC, and 2" the 
highest power of 2 which enters into D, then either 7)i must be of the form 
Sw + 1, or if not, then m must be greater than 3m. 

These three conditions being satisfied, the given equation can always be 
transformed into another, 

A'u' + B'v^ + Cw' = D'uvw, 
where 

A'B'G' = ABG, D' = D, uvw = a factor of z. 

The consequence of this is, as stated in my former paper, that wherever 
A, B, G, D, besides satisfying the conditions above stated, are taken so as 
likewise to satisfy the condition, — firstly, of ABG being equal to 2^™*^ or 
secondly, of ABG being equal to 2^™*^ . p'"*\ provided in the second case 
that ABG is of the form 9m + 1, and that D is divisible by 9, p being in 



20] On the Equation in Numbers Ax^ + By^ + CV = Dxyz. Ill 

both cases a prime, then the given equation will be generally insoluble. And 
I am now enabled to add that the only solution of which it will in any case 
admit, is the solitary one found by making two of the terms Ax^, By^, Cz^ 
equal to one another ; so that, for instance, if the given equation should be of 
the form 

ar* + 2/^ + ABGz^ = Dxyz, 

then the above conditions being satisfied, the one solitary solution of which 
the equation can possibly admit, is a; = 1, 3/ = 1, 

Az^-Dz + 2 = 0, 

which may or may not have possible roots. I call this a solitary or singular 
solution, because it exists alone and no other solution can be deduced from 
it ; whereas in general I shall show that any one solution of the equation 

Ax'^ By^ + Gz" = Dxyz 

can be made to furnish an infinity of other solutions independent of the one 
supposed given, that is, not reducible thereto by expelling a common factor 
from the new system of values of x, y, z deduced from the given system. 

The following is the Theorem of Derivation in question : 

Let 



Then if we write 
and make 



Aa? + 5/3' + Cy = Da^y. 
F=Ad\ G = B^\ H^G'f, 

- X = F"-G + G'H + H"-F - 3FGH, 
y = FG'-+ GH- + HF-' - SFGH, 

2r = 1 [F' + G' + H'<- -iFGH], 
a^y{F^+G' + H-^-FG-FH-GH}, 



we shall have 



x' + f + ABCz^ = Dxyz. 

I am hence enabled to show that whenever a^ + y^ + Az'^ = Dxyz is 
insoluble, there will be a whole family of allied equations equally insoluble. 
For instance, because x^ + y' + z'^= is insoluble in integer numbers, I know 
likewise that 

^6 _(. y6 _,. ^6 _ ^yS ^ ^.3^3 ^ yS^S 

«" + 2/" + a" = x^y^ + x^z^ — ly^z^ 
are each equally insoluble. 



112 On the Equation in Numbers [20 

In fact 

(«^ + y^ + z') X {x'^ -\-y^ + z^ — a?if - oi?z^ — y^z^) 

X («" + 2/° + 2" — ^2/^ ~ ^•^^ + 2^/ V) 
X («" + y^ + z^ — 2/V — y V + 2«V) 
X («" + 2/" + 2" — a?'^^ — z^y'^ + 22/^a^) 
= tt^ + «' + w', 
where ?f, v, w are rational integral functions of x, y, z. 

Hence each of the factors must be incapable of becoming zero*. 

As a particular instance of my general theory of transformation and 
elevation, take the equation 

a? + y^+2z^ = Mccyz. 

Then, with the exception of the singular or solitary solution a; = l, y = l, of 
which I take no account, I am able to affirm that for all values of M between 
7 and — 6, both inclusive, with the exception of ilf = — 2, the equation is 
insoluble in integer numbers. 

Take now the equation where M = — 2, namely 
a-^ + 2/* + 22= + '2.xyz = 0. 

One particular solution of this is 

x=l, y = —l, 0=1. 

Another, which I shall call the second -f, is 

X = 1, y = S, z = — 2. 
From the first solution I can deduce in succession the following : 
a; =11, 2/ = -5. •s = — 7, 

*■ = - 793269121, 2/= 1179490001, 2 = - 1189735855, 
&c. &c. &c. 

From the second, 

« = - 1008.5, 2/ = 8921, 2 = -8442, 
X = &c. y = &c. 2 = &c. 

As another example, take the equation 

af^ + y^ + Qz^ = Qxys. 

* It is however sufficiently evident from their intrinsic form, whioli may be reduced to 
1{M' + SN^), that this impossibility exists for all the factors except the first, 
t See Postscript. 



i 



will be 












x = 


= 17, 


The equation 








admits of the solutions 








X ■■ 


= 1, 






X-- 


= — 


271. 



20] Ax' + By' + Cz' = Dxyz. 1 1 3 

One solution of the transformed equation 

u^ + 2v^ + Sw* = Quvw 
is evidently 

!t = 1, v = l, w = 1. 

Hence I can deduce an infinite sei'ies of solutions of the given equation, of 
which the first in order of ascent will be 

x= 5, y = 7, z = S. 

Again, the lowest possible solution in integers of the equation 

ar' + y' 4- 6^:^ = 

y = 37, ^ = -21. ' 

' + y'' + 92^ = 

2/ = 2, 0=-l, 

2/ = 919, z = -i38. 

I trust that my readers will do me the justice to believe that I am 
in possession of a strict demonstration of all that has been here advanced 
without proof. Certain of the writer's friends on the continent have, in their 
comments upon one of his former papers which appeared in this Magazine, 
complimented his powers of divination at the expense of his judgment, in 
rather gratuitously assuming that the author of the Theory of Elimination 
was unprovided with the demonstrations, which he was too inert or too beset 
with worldly cares and distractions to present to the public in a sufficiently 
digested form. The proof of whatever has been here advanced exists not 
merely as a conception of the author's mind, but fairly drawn out in writing, 
and in a form fit for publication. 

P.S. It must not be supposed that the two primary or basic solutions 
above given of the equation 

x^ + y^ + 2z^ + 2xyz = 0, 
namely, oo=l, y — — l, z =1, 

x = l, y = 3, z = -2, 

are independent of one another. The second may be derived from the first, 
as I shall show in a future communication. In fact there exist three inde- 
pendent processes, by combining which together, one particular solution may 
be made to give rise to an infinite series of infinite series of infinite series of 
correlated solutions, which it may possibly be discovered contain between 
them the general complete solution of the equation 
af + y^ + Az^ = Dxyz. 



21. 



ON THE GENERAL SOLUTION (IN CERTAIN CASES) OF 
THE EQUATION a? + y^ + Az' = Mxyz, &c. 

[Philosophical Magazine, xxxi. (1847), pp. 467 — 471.] 

I SHAiiL restrict the enunciation of the proposition I am about to 
advance to much narrower limits than I believe are necessary to the 
truth, with a view to avoid making any statement which I -may hereafter 
have occasion to modify. Let us then suppose in the equation 

«* + 2/' 4- Az^ = Mxyz 

that J. is a prime number, and that 21 A — M^ is positive, but exempt from 
positive prime factors of the form 6i + 1. Then I say, and have succeeded 
in demonstrating, that all the possible solutions in integer numbers of the 
given equation may be obtained by explicit processes from one particidar 
solution or system of values of x, y, z, which may be called the Primitive 
system. 

This system of roots or of values of x, y, z is that system in which the 
value of the greatest of the three terms x, y, A^ .z (which may be called the 
Dominant) is the least possible of all such dominants. I believe that in 
general the system of the least Dominant is identical with the system of the 
least Content, meaning by the latter term the product of the three terms out 
of which the Dominant is elected. I proceed to show the law of derivation. 

To express this simply, I must premise that I shall have to employ such 
an expression as S' = (j) (S) to indicate, not that a certain quantity, S', is 
a function of S, but that a certain system of quantities disconnected from 
one another, denoted by *S", are severally functions of a certain other system 
of quantities denoted by S; and, as usual, I shall denote cfi^S by ^^S, 
(p^'S by (j}^S, and so forth. 

Let now P be the Primitive system of solution of the equation 

^ a^ + y^ + Az^ = Mxyz, 

P denoting a certain system of values of and written in the order of the 



21] On the Equation oi? + y'' + Az^ = Mxyz, d'c. 115 

letters x, y, z, which may always be found by a limited number of trials 
(provided that the equation admits of any solution). That this is the case is 
obvious, since we have only to give the Dominant every possible value from 
the integer next greatest to ^4* upwards, and combine the values of a?, y^, 
Az^ so that none shall ever exceed at each step the cube of such dominant, 
and we must at last, if there eocist any solution, arrive at the System of the 
Least Dominant. 

Now, every system of solution is of one or the other of two characters. 
Either x and y must be odd and z even, or x and y must be one odd and the 
other even and z odd. That all three should be odd is inconsistent with 
the given conditions as to A being odd and M even ; and if all three were 
even, by driving out the common factor we should revert to one or the other 
of the foregoing cases. 

The systems of solution where z is even may be termed Reducible, those 
where z is odd Irreducible. Let ^ denote a certain symbol of transformation 
hereafter to be explained. 

Then the Reducible systems of the first order may be expressed by 
<^P, <^^P, <j)^P, ad infinitum ; 
or in general by (^"'P, '% being absolutely arbitrary. I will anticipate by 
stating that the function cf) involves no variable constants ; that is to say, 
(/) (S) may be found explicitly from S without any reference to the particular 
equation to which S belongs. Let now yfr denote another symbol of trans- 
formation, also hereafter to be defined, and differing from cj) insofar as it does 
involve as constants the three values of x, y, z contained in P : then the 
general representations of Irreducible systems of the first order will be 
denoted by i}r(j)"'>P. 

It. is proper to state here that the symbol -^jr is ambiguous ; and ■\|r(^"'P, 
when P and n^ are given, will have two values, according to the way in 
which the terms represented by P are compared with x, y, z in the given 
■equation 

«' + y' + Az^ = Mxyz ; 

for it is obvious that if x = a, y = b,z = c satisfies the equation, so likewise 
■will 

x = b, y = a, z= c. 

Each however of these values of ■\lr(j)'^'P gives a solution of the kind above 
designated. 

Proceeding in like manner as before, the Reducible system of the second 
order may be designated by <p'"^ . -v/r^"' . P, the Irreducible by T/r(/>"2 . -v/r^"' . P ; 
and in general every possible system of values of x, y, z satisfying the proposed 
equation, in which z is even, is comprised under the form 



116 On the Greneral Solution of [21 

and every possible system of such values, in which z is odd, is comprised 
under the form 

■xfrcj)''^'' . ■yfrdf'r-l . yjr. . .■'^(^'"■^ . P : 

the quantities % , Wg • • • «r being of course all independent of one another, 
and unlimited in number and value. 

Thus then we may be said to have the general solution of the given 
equation in the same sense as an arbitrary sum of terms, each of a certain 
form, is in certain cases accepted as tlie complete solution of a partial 
differential equation. 

As regards the value of the symbols ■y^ and (/>, ^ indicates the process by 
which a, b, c becomes transformed into a, yS, 7, the relations between the two 
sets of elements being contained in the following equations : 

a' = a^, h' = b^, c' = Ac^, 

a = a'-V + h'-c' + c"^a' — Sa'b'c', 

/3 = a'b'- + b'c'- + c'a"^ - 'ia'b'c', 

ry = abc {a- + b'- + c- — a'b' — a'c' — b'c'}. 

Next, as to the effect of the Duplex symbol -v/r. Let e, g, t be the 
elements of the Primitive system P : 1 being the value of z and e, g of 
X and y taken in either mode of combination, each with each, which satisfy 
the proposed equation 

cc^ -\-y^ + Az' = Mxyz. 

Let I, m, n represent any system 8, 

X, jx, V represent any system ■\fr(S), 

■yjrS has two values, which we may denote by t/t'S, 'yjrS respectively, and 
accentuating the elements X, fi, v accordingly to correspond, we shall have 

\' = Sgm (gl — em) + 'iAm (d - en) - M{giP — eHm), 

fi' = 3ALn{i.in — gl) + '3el (em — gl) — M (eim^ — g^lm), 

< v =Sel {en — d) + '^gni (gn - im) — M (egn'' — iHm) : 

we have then 

■\p-'S = X', fx,', V , 

and in like manner 

'■KJrS = 'X, '//., 'v, 

'■ylrS being derived from i/r'/S' by the mere interchange of e and g one with the 
other. 



21] the Equation x' + y'' + Az^ = Mxyz, &c. 117 

I have stated that every possible solution of the proposed equation comes 
under one or the other of the orders, infinite in number and infinite to the 
power of infinity in variety of degree, above given : this is not strictly true, 
unless we understand that all systems of solution are considered to be 
equivalent which differ only in a multiplier common to all three terms 
of each ; that is to say, which may be rendered identical by the expulsion 
of a common factor. So that ma., m/3, mj as a system is treated as identical 
with a, /8, 7, which of course substantially it is ; and it should be remarked 
that there is nothing to prevent the operations denoted by <^ and i|r intro- 
ducing a common factor into the systems which they serve to generate, and 
the latter in particular will have a strong tendency so to do. 

I believe that this theorem may be extended with scarcely any modifica- 
tion to the case where A, instead of being a prime, is any power of the same, 
and to suppositions still more general. I believe also that, subject to certain 
very limited restrictions, the theorem may prove to apply to the case where 
the determinant '21 A — M'^ becomes negative. 

The peculiarity of this case which distinguishes it from the former, is 
that it admits of all the three variables x, y, z in the equation 

a? ■Vy'^ Az" = Mxyz 

having the same sign, which is impossible when the determinant is positive ; 

or in other words, the curve of the third degree represented by the equation 

M 
F^ -1- X^ -I- 1 = -^^XY (in which I call the coefiScient of XF the character- 

A-^ 

istic), which, as long as the quantity last named is less than 3, is a single 
continuous curve extending on both sides to infinity, as soon as the 
characteristic becomes equal to 3 assumes to itself an isolated point, the 
germ of an oval or closed branch, which continues to swell out (always lying 
apart from the infinite branch) as the characteristic continues indefinitely 
to increase. 

I ought not to omit to call attention to the fact that the theorem above 
detailed is always applicable to the case of the equation 

a? + y'->rAz^= 0, 

when A is any power of a prime number not of the form 6i -f- 1 ; in other 
words, the above always belongs to the class of equations having Monogenous 
solutions, which for the sake of brevity may be termed themselves Mono- 
genous Equations*. 

* Thus the equation a;^ + j/^ + 9z^ = alluded to by Legendre is Monogenous, and the Primitive 
system of solution is a;=l, i/ = 2, z= -1, from which every other possible solution in Integers 
may be deduced. 



118 On the Equation x^ + y^ + Az' = Mxyz, &c. [21 

On the probable existence of such a class of equations I hazarded 
a conjecture at the conclusion of my last communication to this Magazine. 
As I hope shortly to bring out a paper on this subject in a more complete 
form, I shall content myself at this time with merely stating a theorem 
of much importance to the completion of the theory of insoluble and of 
Monogeuous equations of the third degree ; to wit, that the equation in 
integers 

a {a? -\- 'f -^ z^) -V c {a?y + y^z + z'^x + xy^ + yz- + zap) + exyz = 

may always be transformed so as to depend upon the equation 

fw' + gv^ + hiifi = (6a — e) uvw, 

wherein fgh = ae- — (c- + 3a^) e + 9(J^ — 3ac^ — 2cl 

By means of the above theorem, among other and more remarkable 
consequences, we are enabled to give a theory of the irresoluble and 
monogenous cases of the equation 

0(? + y^ + w?z^ = Mxyz, 

when m is some power of 2, or of certain other numbers. 



22. 



ON THE INTERSECTIONS, CONTACTS, AND OTHER CORRE- 
LATIONS OF TWO CONICS EXPRESSED BY INDETER- 
MINATE COORDINATES. 

[Cambridge and Dublin Mathematical Journal, v. (1850), pp. 262 — 282.] 

Let ^7=0, V=0 be two homogeneous equations of tbe second degree 
with real coefficients, between the same three variables ^, ly, ^. 

The direct and most general mode of determining the intersections of 
the conies expressed by these equations would be to make 

a^ +br] 4- c^ = t, 

a'^+b'7) + c'^=u: 

eliminating ^, tj, f between the four equations in which they appear, there 
results a biquadratic equation between t and u. The nature of the inter- 
sections will depend upon the nature of the roots of this biquadratic ; and 
thus the conditions may be expressed analytically, which will represent 
the several cases of all the intersections being real or all imaginary, or one 
pair real and the other imaginary. These analytical conditions will depend 
upon the signs of certain functions of the coefficients of the given and the 
assumed equations being of an assigned character ; my endeavour has been 
to obtain conditions of a character perfectly symmetrical and free from the 
coefficients arbitrarily introduced. 

In this research I have only partially succeeded, but the method 
employed, and some of the collateral results, will, I think, be found of 
sufficient interest to justify their appearance in the pages of this Joiirnal. 

Adopting Mr Cay ley's excellent designation, let the four points of inter- 
section of the two conies be called a quadrangle. This quadrangle will have 
three pairs of sides; the intersections of each pair, from principles of 
analogy, I call the vertices of the quadrangle. Then, inasmuch as the four 



120 On the Correlations of two Conies [22 

sets of ratios ^ : ri : ^, corresponding with the four sets of the ratio t : u, 
must be so related that we may always make 

| = a-6V(-l), |;=c-dV(-l), 

bs b3 

| = a_/3V(-l), |; = 7-SV(-1), 

we may easily draw the following conclusions. 

If all the four points of the quadrangle of intersection are real, the three 
vertices and the three pairs of sides are all real. If only two points of the 
quadrangle are real, one vertex and one of the three pairs of sides will be 
real ; the other two vertices and two pairs of sides being imaginary. If all 
four points of the quadrangle are unreal, one pair of sides will be real 
and the other two pairs imaginary, as in the last case ; but all the three 
vertices will remain real, as in the first case. Hence we have a direct and 
simple criterion for distinguishing the case of mixed intersection from inter- 
section wholly real or wholly imaginary ; namely, that the cubic equation 
of the roots of which the coordinates of the vertices are real linear functions 
shall have a pair of imaginary roots. This is the sole and unequivocal 
condition required. 

The equation in question is, or ought to be, well known to be the deter- 
minant in respect to }, ij, ^ of XU + /mV. In fact, if we write 

C = ap + bv' + c^+ 2a'v^+ 2b' ^^ + 2c'^v, 

F = ap + I3v' + 7^ + 2a'^? + 2/3^? + ^f^V, 

\ f7 4- yaF = (aX + «;t.) P + &c. = ^ r^ + ^T + Cr + 2A'^? + 25'?^ + 2C'^r,> 

the ratios of the coordinates ^, v, fof the vertex oi XU + fxV may easily be 
shown to be identical with 

AB-G-: G'A' - B'B : B'C - A' A, 
and will be real or imaginary as X : /u. is one or the other. 

If then the cubic equation in X : /u,, namely, n (X f7 -|- /^ F) = 0, has a pair of 
imaginary roots, that is, if i=)p(X.C/"-l-/iF) is a positive quantity, the inter- 
sections of U and F are of a mixed kind, that is, the two conies have two 
real points in common. 



22] expressed hy Indeterminate Goordmates. 121 

I may remark here, en passant, that if we form the biquadratic equation 
in t and u, cf) (t, u) = from the equations 

F=0, 

af +br] + c^=t, 

a'^ + b'tj + c'^=u, 

and if any reducing cubic of this equation be P (6, a>) = 0, the determinant 
of P (6, a) must, from what has been shown above, be identical with 
CD CDJXU + fj,Y) multiphed by some squared function of the extraneous 

coefficients 

a, b, G] a, h' , c. 

If aa (\U -\- fjiV) is a negative quantity, it remains to distinguish 
between the cases of the conies intersecting really in four points or not at all. 

The most obvious mode of proceeding to distinguish between purely real 
and purely imaginary intersections would be as follows. Let Xj, /ii; X^, fx^; 
Xs, /Lis, be the three sets of values of \, fj, which satisfy the equation 

and make 

Ai = a\i + a/j,j, -42 = «'^2 + ftMs. -^3 = a\s + a/u.3, 

(7i = cXi + 7/cii , C2 = c\„ + 7/x2, C3 = 0X3 + 7^3, 

B,' = b\ + l3'/xi , Bo' = h'X. + /3>2 , B^ = 6% + /3>3 , 
Afi,-Br- = e„ A,0,-B,'- = e,, A,C,- B,'- = e,. 

Now if the equation 

A^"- + Br + Of = + 2A'rii;+ 2B^^ + 2G'^rj = 

represent a pair of straight lines, it may be thrown into the form 

. „ AG-B'-' „ _ 

Au' + 2 '"' = 0> 

where u and v are linear functions of f, tj, f, and the straight lines will be 
real or imaginary, according as B'- — AC is positive or negative ; hence one or 
else all of the quantities ei, e^, eg, will be necessarily negative, and the inter- 
sections will be all real or all imaginary, according as all three are negative 
or only one is so. A cubic equation in e may be formed containing e^, e.2, 63 
as its roots by eliminating between the equations 

e = AC-B'"-; a (\U + /j.V) = 0, 

and the conditions for the reality of the intersections will be that all four 
coefficients of this cubic shall be of the same sign, which in reality amount 
only to two, since the first and last must in all cases have the same sign. 



122 On the Correlations of ttoo Conies [22 

The same objection however of want of symmetry and consequent 
irrelevancy and complexity attaches to this as much as to the method 
originally proposed. The following treatment of the question relieves the 
objection of want of symmetry as far as the coefficients of the same equation 
are concerned, but in its practical application necessitates an arbitrary and 
therefore unsymmetrical election to be made between the two sets of coeffi- 
cients appei-taining to the two equations. It is however, I think, too curious 
and suggestive to be suppressed. 

I observe that if the four intersections are all real, an imaginary conic 
cannot be drawn through them ; for the equation to an imaginary conic may 
always be reduced to the form Ax' -\- By'^+Gz' = 0, where A, B, G are all 
positive and can therefore have at utmost one real point. Consequently 
the case of total non-intersection is distinguishable from that of complete 
intersection by the peculiarity that in the one case fi may be so taken that 
U+fj,V=0 shall represent an imaginary conic, that is, U + fiV will be a 
function whose sign never changes for real values of ^, t], f, whereas in the 
latter case no value of fi will make U + fJ^V=0 the equation to an imaginary 
conic, and therefore U + f^V will have values on both sides of zero. On the 
other hand, it is obvious that an infinite number of real as well as unreal 
conies may be drawn through four imaginary points of intersection. Con- 
sequently if we make U+nV=0 (supposing the intersections of U and V 
to be imaginary), there will be a range or ranges of values of fju consistent, 
and another range or ranges of values of /x inconsistent with real values of 
^, 77, ^; in other words, U±fiV=0 treated as au equation between the four 
variables f, tj, f, fi, will give one or more maxima or minima values of fx, 
in the case supposed, but no such values when the intersections are two or 
all of them real. 

To determine these values of fi, let clfx = Qi; then we have 

|(f^-^F) = 0, 

that is ci^([7-yu,F) = 0. 

In order that any value of /u. found from this equation may be a maximum 
or minimum, Lagrange's condition requires that 

It d , d J dV 
may be a function of unchangeable sign. 



22] expressed by Indeterminate Coordinates. 123 

dU dV ^^dfi 

^°" d^=''d^ + ^dr 

therefore since d/x = 0, 





d'U 


-f- 










-^Gf" 


7-/. 


.F}; 


d 
d^ 


d _ 
' dij 


1 f^ c^ 

Vd^- dv 


[U- 


/^■f^}. 




&c. 


&c. 


&c. 





Hence 



similarly 



Making now as before 

U=a^'-+btj'~ + &c., 

a — /j,a = A, b—fj,0 — B, &c., 

the condition for fi, a root of n {U — ij,V} = 0, giving fi a maximum or mini- 
mum, may be expressed by saying that 

Ah'' + Bk^ + CT + 2A'M + 2B'hl + 2G'hk 

shall be unchangeable in sign for all real values of h, k, I. 

The above quantity, by virtue of the equation n = 0, is always the 
product of two linear functions. Hence we see, as above indicated, that if 
all these pairs are real, that is, if all the points of intersection of U and V 
are real, there is no maximum or minimum value of /j, ; but if only one pair 
be real and the other two pairs be imaginary, that is, if all the four inter- 
sections are imaginary, then two of the values of fi, namely those correspond- 
ing to the imaginary pairs, are real maxima or minima values of /j,, but the 
third is illusory. 

Now I shall show that if F=0 is a real conic, but the intersections of 
U and V are all unreal, the value of /i which makes U + fiV the product of 
real linear functions of f, rj, t, is always one or the other extreme of the three 
values of /i which satisfy the equation 

Assume as the three axes of coordinates the three lines joining the 
vertices of the quadrangle each with each, the two non-intersecting conies 
may evidently be written under the form 

U= c {a? + y~)- e {y- + z") = 0, 



124 On the Correlations of two Conies [22 

these equations being only other modes of writing 

V= A'x- 4- B'y"- + C'z\ 

in which A, B, G; A', B', C" will be real, because by hj^pothesis o{U+fiV)=0 
has all its roots real. 

Hence x, y, z are linear functions of ^, t], ^, and consequently, by a simple 
inference from a theorem of Prof. Boole*, the roots of affy+itF! are 
identical with those of 

These latter are evidently -, -, ; the third of which is the one 

7 6 7 — e 

which makes U + ^uFthe product of two i^eal linears, for we have 
yll + cV=(ce —rye)(y- + z"), 
eU+ eV— (ec — ey) (x- + y-), 
{y-€) U+{c-e) V= (ce - ey) (z- - w-)f. 



Now 



c c — e _ ey — ce 

7 7 - « 7 (7 - e) ' 

e c — e _ 67 — ce _ 

e y — e e (7 — e) ' 

and e, 7 are supposed to have the same sign, as otherwise V would be an 
unreal conic ; hence the ascending or descending order of magnitudes of 

the three values of X follows the scale - , - , , as was to be shown. 

7 6 7 — e 

Imagine now lengths reckoned on a line corresponding to all values of 
/J, from — GO to + CO , and mark off upon this line by the letters A, B, G, 
the lengths corresponding with the three roots of o {U + fiV) = 0. Then 
observing that when yu. = ± 00, U+fj.V is of the same nature as V, and is 
therefore a possible conic by hypothesis, and agreeing to understand by a 
possible and impossible region of /i, a range of values for which U + fxV 
corresponds to a possible and impossible conic respectively, one or the other of 
the annexed schemes will represent the circumstances of the case supposed : 



- CO -^ Poss. Reg. 


A Imposs. Reg. B Poss. Reg. 


C Poss. Reg. i^° + 00 


- 00 "^M Poss. Reg. 


A Poss. Reg. B Imposs. Reg. 


1 

CPoss. Reg. ^r + o) 



But in either scheme it is essential to observe that the middle root of 
a {U + fjiV) = divides a possible from an impossible region; and therefore 

* See Postscript. + 2'^-. t^ = of course represents a jvnZ pair of lines. 



22] expressed hy Indeterminate Coordinates. 125 

if we can find n, v, any two values lying between the first and second and 
second and third roots of the above equation arranged in order of their 
magnitude, one of the two equations U+ vV=^0, U+nV=0, will represent 
a possible and the other an impossible conic : one such couple of values 
may always be found by taking the roots of the quadratic equation 

Hence calling the two roots thereof in and M, we see (which is in itself 
a theorem) that one at least of the conies U+mV=0, U + MV=0, must 
be a possible conic, provided only that V=0 be a possible conic: if both 
U + mV and U+MV are possible conies, the intersections of U and Fare 
all real, and if not, not*. The criteria for distinguishing possible from 
impossible conies being well known need not be stated in this place. 

We may of course proceed analogously by forming the two conies IJJ + V, 

d 
LU +V, where I and L are roots of 3- a {'\.U+ F} = upon the supposition 

oi U=0 being a possible conic. 

If either of the two U and F be not possible, their intersections are of 
course impossible, and the question is already decided. 

It will be seen as pre-indicated that this method only fails in symmetry 
because of the choice between the couples m, M, and I, L. But moreover 
a perfect method for the discrimination of the two cases of unmixed inter- 
section one from the other should (perhaps ?) require the application of only 
a single test (in lieu of the two conditions which the above method supposes), 
over and above the condition which expresses the fact of the intersections 
being so unmixed. Such more perfect method I have not yet been able 
to achieve. 

Another interesting question of intersections remains to be. discussed, 
namely, supposing the two conies are known to be non-intersecting, how are we 
to ascertain if they are external to one another, or if one contains the other? 
In order to settle this point we must first establish a criterion for determin- 
ing whether a given point is internal or external to a given conic ; the point 
being in general said to be external when two real tangents can be drawn 
from it to the curve, and internal when this cannot be done. 

* It must be \vell observed however that the possibility of the conies P+niF and of TJ+MV 
does not imply the reality of the intersections unless the conic V is known to be possible. 

For if V be impossible c and 7 have opposite signs, and therefore -— - is intermediate between 

- and - , and the scheme for ^ will be as here annexed : 



so that U+mV and U+MV will both represent possible conies. 



126 



On the Correlations of two Conies 



[22 



Let now 

<^ (a;, y, z) = aaf' + by- + cz" + 2a' yz + 2b' zx + 2c'ii:y = 0, 

be the equation to any conic ; I, m, n the coordinates of any point. Let 
A =bc — a'-, B = ca — b'-, C = ab — c'-, 
A' = cm' — b'c, B' = bb' — c'a', C = cc — a'b'. 

Then the reciprocal equation to the conic is 

A^- + Br + C!r- + 2A'r,^+ 25'?f + 2C"f r; = 0, 

and in making l^ + m7j+ni^=0, the ratios of f, t), f must be real if the 

tangents drawn from I, m, n are real : this will be found to imply that the 

determinant 



A, C, B', 

C, B, A', 

B', A', C, 

l, m, n. 



I 







shall be negative*. This determinant may be shown f to be equal to the 
product of the determinant 

a, c', V 

c, b, a' 

b', a, c 
hy the quantity 

id- + bm- + en- — 2a' inn — 2b'ln — 2c Im, 
that is, equal to (fy (I, m, n) x a. 

Hence I, vi, n is internal or external to <p (o:, y, z) according as (^ {I, m, n) 
and Qc^ have the same or contrary sign. 

If {I, m, n) = 0, the point lies on the conic, and the point is neither 
internal nor external ; if D (^ = 0, the conic becomes a pair of straight lines, 
and no point can be said either to be within or without such a system. 
Hence our criterion fails, as it ought to do, just in the very two cases where 
the distinction vanishes. I believe that this criterion is here given for 
the first time. 

* See theorem of the " Diminished Determinant " in Postscript to this paper. 

+ As we know a priori hy ■virtue of a theorem given by M. Canchy, and which is included as 
a particular case in a theorem of my own, relating to Compound Determinants, that is, Deter- 
minants of Determinants, which will take its place as an immediate consequence of my funda- 
mental Theorem given in a Memoir about to appear. The well-known rule for the multipUeation 
of Determinants is also a direct and simple consequence from my theorem on Compound 
Determinants, which indeed comprises, I believe, in one glance, all the heretofore existing 
Doctrine of Determinants. 



22] expressed hy Indeterminate Coordinates. 127 

To return to the two non-intersecting conies. Let us again throw them 
under the form 

U-={x'^ + y"-)-e'{z''-\-y-), 

V = k {a? + y-) — ke^ (z^ + y"), 

e and e being real, that is, U and V being both functions corresponding to 
possible conies. Suppose U external to V; then any point in TJ is an 
external point to V. 

Take in U either of the two points represented by the equations y = 0, 
ap- = e-z- ; substituting these values of y and x, V becomes k (e- — e^) z", 
and oV becomes — k^e^ {1 — e^) ; therefore (1 — e^) (e^ — e^) must be positive, 
that is, e' must be one of the extremes of the three values 1, e^, e^. In like 
manner, if V is external to U, e will be also one of the extremes of the same 
three quantities ; and hence, if the two conies are mutually external, unity 
will be the middle magnitude of the group eP, 1, e". 

Now the three roots oi a(V+XU) = 0, are 

' e- ' 1 — e^ ' 

Hence if U and V be without one another, or, as it may be termed, are 
extra-spatial, the third value of X will be of a different sign from the first 
two ; but if the two conies be co-spatial, that is, if one includes the other, all 
the three values of X will have the same sign. Hence we have the following 
elegant criterion of co-spatiality of two po.^sible conies expressed by the 
equations U=0, V=0, between indeterminate coordinates ^, tj, ^; the 
coefficients of the cubic function p (XZ7-f-yu.F) must give only changes or 
only continuations of sign. 

If this test be not satisfied, it will remain to determine which of the two 
conies contains, and which is contained by the other. Let U contain V, 

1 _ g2 

then the order of magnitudes will be 1, e", e^ ; therefore k „ is greater 

1 - e- 

than k, and therefore k- „, which is that root of the equation a{V+XU) = 

which is always one or the other of the extremes, is the greatest of the three. 
Hence the scheme for the impossible and possible regions of X will be as 
below : 

- °o "^Si Poss. A Imposs. B Poss. C Poss. ^W" + oo 



Hence if the two roots oi -j- {V + XU} = be I and L, and of the two 

conies V+IU = 0, V+ LU=0, the former be the possible, and the latter the 
impossible one, U contains V or is contained in it according as I is greater 
or less than L. 



128 On the Correlations of two Conies [22 

Observe that if U and V be non-cospatial, so that the three values of 
fj, in D(U+fiV) = have not all the same sign and consequently zero lies 
between the greatest and least of them, it will not be necessary to make 
trial of the characters of the two curves U+ 'mV=0, and U+MV=0, 
in order to ascertain whether U and V intersect or not ; for it will be 
sufficient to find which of the two quantities m and M substituted for /j, 
in o{U + /mV) causes it to have the opposite sign to a (U + 0V), that is, 
aU, and this one of the two it is, if either, which will make U + fj.V an 
impossible conic, and will thus alone serve to determine whether the inter- 
sections of U and V are unreal, or the contrary. 

It might be a curious question to consider whether, in a certain sense, 
conies not both possible may not be said to lie one within or without the 
other. Upon general logical grounds, I think it not improbable that two 
impossible conies might be discovered each to contain the other; but this is 
an inquiry which I have not had leisure to enter upon. 

I have thus far supposed the roots of o{XU -{■¥)= to be all distinct 
from one another. I now approach the discussion of the contact of two 
conies, in which event two or more of the roots will be equal. The condition 
for simple contact is evidently t=i p (X C/ + /i F) = 0. 

The unpaired value of \ in n(\U + V) makes XU + V an impossible 
pair of lines, and therefore, in the scheme for X drawn as above, will separate 
the possible from the impossible region. 

Whether the conies intersect in two real or two unreal points, besides the 
point of contact, will be known at once by ascertaining whether V +fj,V=0 
represents two real or two imaginary lines. If the latter, the two curves lie 
dos-d-dos or one within the other, according as the successions of sign in 
D (X.U +V) are all of the same kind or not ; if they be all of the same kind, 
one will include the other, namely, U will include V if the equal roots are 
greater, and be included in it if they be less than the unequal one. This last 
conclusion however, it should be observed, is inferred upon the principle of 
continuity, by making two values of X approach indefinitely near to one 
another, but cannot be strictly deduced from the equations given for U and 
F applicable to the general case, in which the axes of coordinates are the' 
three axes joining the vertices ; since these latter, in the case supposed, 
reduce to two only, and consequently such representation of U and F 
becomes illusory. 

If all three values of X are equal, the three vertices come together, 
and hence the two conies will have three consecutive points in common, 
that is, will have the same circle of curvature. On this supposition the two 
curves cut at the point of contact, and all four points of intersection are 
of course real. 



22] expressed by Indeterminate Coordinates. 129 

The classification of contacts between two conies maybe stated as follows: 

Simple contact = one case. 

Second degree contact = two cases, namel_y, common curvature or double 
contact. 

Third degree contact = one case, namely, contact in four consecutive points. 

These four cases of course correspond to the several suppositions of there 
being two equal roots, three equal roots, two pairs of equal roots, or four 
equal roots in the biquadratic equation obtained between two variables by 
elimination performed in any manner between the given equations in the 
two conies. 

The first species and the first case of the second species have been already 
disposed of I proceed to assign the conditions appertaining to the second 
case of the second species, when U and V have a double contact. 

Let A, A', B, B' be the two pairs of coincident points in which the 
conies are supposed to meet; either pair of lines AB, A'B', and AB', A'B, 
becomes a coincident pair. Hence such a value of /i can be found as will 
niake U+ fiV the square of a linear function- of ^, rj, ^. If therefore we 
make 17+ /j,V= W, and form the determinant 
d'W d'W d'W 

d^dv' d^d^' ^ 

d^W d'W 

drf ' drjd^' 

d?W d}W_ 
d^dv' d^^ ' 



drW^ 
dTjd^' 

d'W 
d^d^' 







= Af + Bq^ + Gr"- + 2Fqr + 2Grp + 2Hpq, 
where all the coefficients are quadratic functions of fj,, and make 

^ = 0, 5 = 0, (7 = 0, F=0, G = 0, H = 0, 
each of these six equations in fi will have one and the same root in common. 

It is, however, enough to select any three ; if these vanish together 
for any value of /m, the remaining three must also vanish. This is a simple 
application of a general law * which will appear in a forthcoming memoir on 
"Determinants and Quadratic Forms," of which this paper is to be considered 
as an accidental episode. 

* For statement of this law called the Homaloidal Law, see Philosophical Magazine of this 
month "On Certain Additions, &c." [p. 150 below. Ed.] 

s. 9 



130 On the Correlations of two Conies [22 

Take now any three of the six equations which for the sake of generality 
call P = 0, Q = 0, R = 0. The hypothesis of double contact requires that 
P and Q, Q and R, R and P shall have a factor in common ; but these 
conditions are not sufficiently explicit for our present object, since P, Q, R 
might be of the form 

K{\ — a){\ — b), K (A, — b){\ — c), k" (X — c) (A, — a), 

and would thus satisfy the conditions above stated, without P, Q, R having 
a common factor. A sufficient criterion is that fQ + gR and P shall have 
a common factor for all values of/ and g. 

Let then the resultant oifQ + gR and P be 
Lp + Mfg + Nf, 
we must have 

i=0, M=Q, i\^=0, 

where L is the resultant of P and Q, 

N „ „ „ P and Q ; 

and M is a new function, which if we call Q = <p (A.), R = '^ (X), and suppose 
a and h to be the two roots of P = 0, is easily seen to be equal to 

^a . i^h + j)h . i/ra. 

This I call the connective oi P . Q and P . R. 

L, M, N may conveniently be denoted by the forms 

P.Q, P.R, Q.P.R. 

We may now take more generally 

aP + bQ + cR, 

aP+l3Q + jR, 

which will have a factor in common for all values of a, b, c, a, /3, y. 

I am indebted to Mr Cayley for the remark that the resultant of these 
two functions is a new qiiadratic function, which, according to my notation 
just given, may be put under the form 

PQ {a/3 - baf + QR {by - c/Sf + RP {ca - ay)' 

+ PRQ {by - c/3) {col- ay) + QPR {ca - ay){a^ - boc) + RQP {a^ - ba){by - c/3). 

Ternary systems of the six coefficients formed upon the type of {PQ, 
PQR, QR), I call complete systems, because the three functions included in 
such a system equated severally to zero, imply that the remaining three 
coefficients are all zero. Such a system as {PQ, QR, RP) I term an incom- 
plete ternary system as not drawing with it the like implication. Probably (?) 
we should find on investigation that PRQ, QPR, RQP, would also be an 



22] expressed by Indeterminate Coordinates. 131 

incomplete system, but that systems formed after the type of PRQ, RQ, RQP 
are complete. This however is only matter of conjecture, as I have been too 
much occupied with other things to enter upon the inquiry. The distinct 
types of ternary systems are altogether six in number, namely, four of a 
symmetrical species, 

FQ, QR, RP, 

PRQ, QPR, RQP, 

PQ, PQR, QR, 
PRQ, RQ, RQP; 
and two of an unsymrnetrical species, namely, 

PQ, PQR, PR, 
PRQ, RQ, QPR* 

If instead of confining ourselves to three out of the six original quantities, 
A, B, C; F, G, H, we take them all into account, and write down the 
resultant of 

aA+hB + cG +fF + gG + hH, 

aA+l3B + ryC+<j>F + xG + vH; 

we shall obtain a quadratic function of 15 variables (not however all indepen- 
dent) having 120 coefficients, all of which must be zero. It would be 
extremely interesting to determine how many complete ternary groups can 
be formed out of these 120 terms. 

It will be recollected that we have assigned as the condition of contact 
in three consecutive points, that a certain cubic equation shall have all its 
roots real. Now, as well remarked by Mr Cayley, we cannot express this 
fact by less than three equations in integral terms of the coefficients. Thus 
if the cubic be written 

aX' + SbX' + :3cX + d = 0, 

we have as one of such ternary systems, 

U=ac-¥ = 0, F=6(i-c- = 0, W=bc-ad = 0. 

The significant parts of these equations are of course, however, capable of 
being connected by integral multipliers U', V, W, such that 

VU+V'V+W'W=0. 

* FQ, QR, SP, may be compared in a general way with the angles, and PRQ, QPP, RQP, 
•with the sides of a triangle. 

9—2 



132 On the Correlations of two Conies [22 

Any number of functions U, V, W so related, I call syzygetic functions, 
and U', V, W I term the syzygetic imdtipliers*. These in the case 
supposed are c, a, b, respectively. 

In like manner it is evident that the members of any group of functions, 
more than two in number, whose nullity is implied in the relation of double 
contact, whether such group form a complete system or not, must be in 

syzygy- 

Thus FQ, PQR, QR, must form a syzj'gy; nor is there any difficulty 
in assigning a system of multipliers to exhibit such syzygy. Galling 
P=(f>{X), R = -\jr {X), a and b the two roots of Q = 0, I have found that 

{(■fay + iirby-} PQ - {4>a . -.|ra + ^6 . -«|r&) PQR + [{4>ay + {(j)by} QR = 0. 

Again, if we take the incomplete system 

{PQ), {QR), {RP), 
it will be found that 

L {QR) -}- M {RP) + N {PQ) = 0, 

provided that, calling a, b; c, d; e, f, the roots of P=0, Q = 0, R=0, 
respectively, we make 

L = {h + ha + k,a' + k.a^ + k.a^) {a- c){a-d){a^- e){^-f) 

+ {k, + hb + hb^ + hb-^ + h¥) {hj-c)JJ >-d){b^ -e)Q>-f) ^ 
M= {k, + he + hc^ + k.c^ + hc^) { o-a){o-b){c-J) {c-e){e-f) 

+ {h + hd + hdr + hd^ + kj^) {d-a){d-b){d^-c){d-e){d-f) ^ 
N = {k, + he + k..e^ + k,e^ + k,e^) ^"^^ ^' " ^^ ^^ " '^ ^' ' ^^ 



+ (^-0 + kj+ hp + k,p + hp) 



e-f 
{f-a){f-b){f-c){f-d)_ 



ko, h, h, h, ki being quite arbitrary, and L, M, N, although presented in 
a fractional form, being essentially integral. 

This fact of L, M, N constituting a system of multipliers to the syzygy 
QR, RP, PQ, is easily demonstrated ; for 

QR = {c-e){c-f){d-e){d-f), 

RP = {e-a){e-b){f-a){f-b), 

PQ = {a- c) {a -d){b- c) {b - d). 

* There will be in general various such systems of multipliers. 



I 



22] exjjressed by Indeterminate Coordinates. 133 

Hence L{QR) + M (RP) + N (PQ) 

= (a - c)(a -d)(a- e)(a -f){h - c){b - d)(b - e)(b-f){c-e){c-f){d-e)(d-f) 

^ ko + k^a + ha- + Icstt^ + k^a' _ 
(a — b){a — c){a — d){a — e){a—f) 

My theory of elimination enables me to explain exactly the nature of 
L, M, iV, and the reason of their appearance as syzygetic factors. 

Let L,-, Mr, Nr signify what L, M, N become, when all the k's except k,. 

are taken zero. Then the theory given by rne in the Philosophical Magazine 

for the year 1838, or thereabontsf, shows that L„\ + Li is the prime derivee 

of the first degree between the two equations P. and Q x R, or, in other 

OR 
words, will be the remainder integralized of —p . 

In like manner M^X + M^, i\^o^ + -^i are the integralized remainders of 
— Y and of -^ respectively. 

If now the resultant of P, Q and of Q, R are each zero, but the resultant 
of P and R is not zero, it will be evident that P, Q, R must be of the form 

/(X + a) (X + c), g{\ + c)(\ + d), h (X + d){\ + b); 

and therefore P x R will contain Q, and consequently we must have 

M„ = 0, M, = 0. 

More generally, if we write 

Q = 0, 

\Q = 0, 

X-Q = 0, 

PxiJ=0, 

and eliminate dialytically, that is, treating X^ X^ X^ X as distinct quantities, 
we shall obtain * 

X^ : X» : X= : X : 1 : : ifi : ifs : M, : Jlfi : if„ ; 
and therefore when P x R contains Q, 

Mo = 0, M, = 0, M, = 0, M^ = 0, M, = 0. 

* This cannot be obtained directly from what is stated in the paper referred to, although 
contained in the general theory of derivation there given. The arbitrary functions which enter 
into the expression for the general derivees have been in that paper evaluated only for the prime 
derivees, which however are only particular phenomena, with reference to the general results of 
Dialytic EHmination. Hereafter I may give a more general exposition of this remarkable, 
although ignored or neglected theory. The prime derivees of fx and f'x are Sturm's Functions, 
cleared of quadratic factors, and are expressed by virtue of the general theorems there laid down 
as functions of x and of symmetrical functions of the roots of fx. [t p- 40 above. Ed.] 



134 On the Correlations of tivo Conies [22 

In like manner, when Q x P contains R, 

and when R x Q contains P, 

Accordingly, we see from the equation 

L (QR) + M {RP) + N {PQ) = 0, 
that iiQR = 0, RP = 0; but PQ not = 0, then N=Q; and therefore 

and so in like manner for the remaining corresponding two suppositions*. 

Before proceeding to consider the remaining case of the highest species 
of contact, I must observe that besides the equations involved in the condi- 
tion that A, B, C; F, Q, H, or, which is the same thing, that any three 
of them shall all have a factor in common, we must have D (fJ + XF) con- 
taining the square of such common factor. In the memoir before adverted 
to a general theorem will be given and proved, which shows that this latter 
condition is involved in the former one ; in fact, more generally (but still 
only as a particular case) that when U and V are quadratic functions of n 
letters, but U+eV admits of being represented as a complete function of 
(?i — 2) quantities only, which are themselves linear functions of the n letters, 
then n{U + XV), which is of course a function of X, of the nth degree, will 
contain the factor (X — e)-. 

When the two conies have four consecutive points in common, the 
characters of double-point contact and of contact in three consecutive points 
must exist simultaneously; and consequently the factor common to A, B, C; 
F, G, H, will enter not as a binary but as a ternary factor into n(f/"-f-XF). 
This gives the extra condition required. As an example take the two conies, 

V = f + x'- 2kxz + (-Ik -l)z^ = 0, 
U + \V= ("— -r -F x] y--F(l -1- X) of - {1 +X(1 - 2k)} z" - 2h\xz. 

* Since we are able to assign the values ot the syzygetie multipliers in the equations 

L (PQ)+ilf {QB) +N {RP] = 0, 

L' {PQ) + M' {PQR) + N' {QR) = 0, 

L" {QE) + M" {QSP) + N" {RP) = 0, 

L'" (RP) + 21'" (RPQ) + N'" (PQ) = 0, 
it follows that we may eliminate between these four equations any three of the six quantities 
(PQ), (PRQ), &c., and thus express any one of them in terms of any two others: this method, 
however, is not practically convenient. I may probably hereafter return to this subject. 



22] expressed by Indeterminate Coordinates. 135 

The complete determinant of ?/ + \F is then 
Y~k (1 + (1 - ^) ^1 {(1 + ^y - 2^-^ (1 + A,) + A;^=} = - j4^ (1 + (1 - yt) \]\ 

A, B, C are the determinants of U+\V, when x=0,y = 0, z = 0, respectively. 
Thus 

^=(izr/, + ^)(i + '^)' 

G = k'X' - (1 + X) (1 + X (1 - 2^-)} = \- (1 - k)- -2\{l-k)-l; 

X = — , — r makes A=0, B=0, 0=0, and the factor X + :r — r enters cubed 
1—k 1—k 

into a (U + W). 

Hence the two conies have a contact of the third order. 

This is easily verified ; for if we pass from general to Cartesian and 
rectangular coordinates, and make z unity; f7=0 will represent an ellipse 
with centre at the origin, eccentricity f/k, and mean focal distance 1, and 
V=0 the circle of curvature at the extremity of the axis major*. 

I had intended to have added some other remarks connected with the 
present discussion, and also to have appended an d posteriori proof of the 
propositions relative to the reality and otherwise of the vertices and chordal 
pairs of intersection which I have, at the commencement of this paper, 
deduced quite legitimately, but in a manner not at first sight perhaps easily 
intelligible, from the general principles of conjugate forms ; but this dis- 
cussion has run on" already to a length so much greater than I had 
anticipated and than the importance of the inquiry may seem to justify, 
that I must reserve for a future number of the Journal what further matter 
I may have to communicate concerning it. 

Postscript. — As I have alluded to Professor Boole's theorem relative to Linear 
Transformations, it may be proper to mention my theorem on the subject, which 
is of a much more general character, and includes Mr Boole's (so far as it refers to 
Quadratic Functions) as a corollary to a particular case. The demonstration will 
be given in the forthcoming memoir above alluded to. 

Let f7 be a quadratic function of any number of letters x^, Xr^.-.x^, and let 
any number r of linear equations of the general form 

ia,.x-i + „arX.2+ + jfl^Xji^ = 0, 

* We have thus discussed all the four oases of blconical contact: for an exactly parallel 
discussion of the theory of contact of a plane with the curve of double curvature in which two 
surfaces of the second order intersect, see the paper in the Pliilosopliical Magazine for this month, 
before referred to. [p. 148 below. Ed.] 



136 



On the Correlations of txoo Conies 



[22 



be instituted between them : and by means of these equations let U be expressed 

as a function of any {n — r) of the given letters, say of a;,.+i, x,.^^ ^n> ^i^cl let 

U, so expressed, be called M. Let 

■flyX^ + oa,.X'o + + Tfir^n 

be called X,.. Then the determinant of J/ in respect to the {ii — r) letters above 
given is equal to the determinant of 

U + L■^x„+■^ + LnX^^^ + + Z^a;„+,., 

considered as a function of the {n + r) letters 



divided by the square of the determinant 



1^2 J 2'^2 •• 



This I call the theorem of Diminished Determinants. 

If now we have U a function of r letters, and V of r other letters, and V is 
dei'ived from U by linear transformations, that is, by r equations connecting the 
27- letters ; then, since U may be considered as a function of all the 2r letters with 
abortive coeiEcients for all the terms where any of the second set of r letters enter, 
we may apply our theorem of diminished determinants to the question so con- 
sidered, and the result may be found to represent Mr Boole's theorem in a form 
rather more general and symmetrical, but substantially identical with that given 
by Mr Boole. 

Thus suppose \aa? + hxy + \c'if- say P, and \ai^ + ^uv + Jyw^ say Q, are mutually 
transformable by virtue of the linear equations 

Ix + my = Am + ix.v, 

I'x + miy = \'u + fji'v, 

P may be considered as a function of x, y, u, v, and Q as the value of P, when we 

eliminate x and y by virtue of the two linear equations 

Li = lx + my — \u — jj-v = 0, 

Zg = I'x + my — X'u — jxv = ; 

we have therefore by our theorem the determinant of Q equal to the squared 

reciprocal of the determinant ' , multiplied by the determinant 



a. 


b, 


0, 


0, 


I, 


I' 


b, 


"i 


0, 


0, 


m, 


m' 


0, 


0, 


0, 


0, 


-\ 


-X 


0, 


0, 


0, 


0, 


-M> 


-H-' 


I, 


TO, 


-A, 


-/^> 


0, 





I', 


m', 


-V, 


-/^'> 


0, 






22] exjiressed hy Indeterminate Coordinates. 137 

which last determinant is evidently equal to the determinant of P multiplied by 
the square of the determinant \ ' ,\. Wlience we see that the determinant of Q 

divided by the square of ' , , is equal to the determinant of P divided by 

the squai'e of I ' , . There is also another way more simple, but less direct, by 

means of which the theorem of diminished determinants may be made to yield 
Mr Boole's theorem of transformation*. Some unavowed use has been made 
in the foregoing pages of this former theorem, one of the highest importance in 
the analytical and geometrical theory of quadratic functions. It has been nearly 
a year in my possession, and I trust and believe that I am committing no act of 
involuntary misappropriation in announcing it as a result of my own researches. 

* Namely, by considering P and Q as each derived from some common function of x, y, u, v, 
w, by means of the equations Lj = 0, L„ = ; the law of Diminished Determinants •will then indicate 
the determinants of P and Q, each under the form of fractions having the same numerator, but 

whose denominators will be I ! , and „' , respectively. 



23. 



AN INSTANTANEOUS DEMONSTRATION OF PASCAL'S 
THEOREM BY THE METHOD OF INDETERMINATE 
COORDINATES. 

[Philosophical Magazine, xxxvil. (1850), p. 212.] 

The new analytical geometry consists essentially of two parts — the one 
determinate, the other indeterminate. 

The determinate analysis comprehends that class of questions in which 
it is necessary to assume independent linear coordinates, or else to take 
cognizance of the equations by which they are connected if they are not 
independent. The indeterminate analysis assumes at will any number of 
coordinates, and leaves the relations which connect them more or less 
indefinite, and reasons chiefly through the medium of the general pro- 
perties of algebraic forms, and their correspondencies with the objects of 
geometrical speculation. Pascal's theorem of the mystic hexagon, and the 
annexed demonstration of its fundamental property, belong to this branch of 
the subject, and afford an instructive and striking example of the application 
of the pure method of indeterminate coordinates. 

Let X, y, z, t, u, v be the sides of a hexagon inscribed in the conic U. Let 
the hexagon be divided by a new line (f) in any manner into two quadri- 
laterals, say xyzcj), tuv(p. 

Then ay<j} + boaz =17= au^ + ^tv ; 

therefore (ay — au) <p = ^tv — hxz ; 

therefore ay — au and <p are the diagonals of the quadrilateral txvz. 

By construction, (f> is the diagonal joining x, v (that is, the intersection of 
X and v) with z, t ; and thus we see that ay — au is the line joining t, x with 
V, z ; but this line passes through y, u. Therefore x,t; y, u ; z, v lie in one 
and the same right line. Q.E.D. 



24. 



ON A NEW CLASS OF THEOREMS IN ELIMINATION 
BETWEEN QUADRATIC FUNCTIONS. 

[Philosophical Magazine, xxxvii. (1850), pjD. 213 — 218.] 

In a forthcoming memoir on determinants and quadratic functions, I have 
demonstrated the following remarkable theorem as a particular case of one 
much more general, also there given and demonstrated. 

Let U and V be respectively quadratic functions of the same 2n letters, 
and let it be supposed possible to institute ii such linear equations between 
these letters as shall make U and V both simultaneously become identically 
zero*. Then the determinant of \U + fjuV, which is of course a function of 
X and /A of the 2?jth degree, will become the square of a function of X and fi 
of the «th degree ; and conversely, if this determinant be a perfect square, XT 
and V may be made to vanish simultaneously by the institution of n linear 
equations between the 2?i letters f. 

Let now P and Q be respectively quadratic functions of three letters only,, 
say X, y, z; and let 

U=P + {lx + my + nz) t, 

V=Q + k{lx + my + nz) t. 

The determinant of XlT+ fiV in respect to x, y, z, t is easily seen to be 
(X + kfjCf X the determinant of 

XP + /xQ + {Ix + my + nz) t 

in respect to x, y, z, t. Hence if we call 

XP + /iQ + {Ix + my + nz) t = W, 

and make cd TF a squared function of X, jjl or which is the same thing, if 

Afi xuzt ' ' 

* In the more general theorem above alluded to, the number of letters is any number m, the- 

number of linear equations being any number not exceeding — . 

t When ?!. = !, we obtain a theorem of elimination between two quadratics, which has been 
already given by Professor Boole. 



140 



On a new Class of Theoi-ems 



[24 



TJ and V will vanish simultaneously when two linear relations are instituted 
between the quantities (all or some of them) x, y, z, t. 

In order that this may be the case, it will be seen to be sufficient that 

P = 0, Q = 0, {lx + my + nz) = 0, 

shall coexist; for then two equations between x, y, z of which lx-\-my-\-nz = ^ 
will be one, will suffice to make TJ and V each identically zero. Hence we 
have the following theorem : 

□ □ f\ f/- + /iF+ {Ix + my + nz) t\ 

is a factor of the resultant of 

P = 0, Q = 0, lx + my + nz=^0. 

A comparison of the orders of the resultant and the determinant shows 
that they must be identical, d-ci-pres, of a numerical factor, which, if the 
resultant be taken in its general lowest terms, may no doubt be easily shown 
to be unity. 

As an illustration of our theorem, let 

P = xy + yz + zx, 

Q = cxy + ayz + hsx. 

0, X + Cfi, X + hfjL, I 
\+ Cfi, 0, X + a/u,, m 
X + 6/i, X + a/jL, 0, n 

1, m, n, 
= n- (X + Cfi)- + m-(X + b/j,f + l'-(X + a/j.)" 

— 2lm (X + b/j.) (X + cifi) — 2mn (X + Cfi) (X + b/j.) — 2nl (X + a/x) (X + Cfi) 
= X- {n- + nv' + I- — 2lm — 2mn — 2nl} 

+ 2X/i [en' + bm- + aP — Im (a + 6) — mn (6 + c) — til (c + a)] 
+ /Li- [c-n- + b-m^ + a-P — 2ablm — 2bcmn — 2canl]. 

And we thus obtain, finally, 

□ n [xP + fiQ + {lx + my + nz)t} 

= (n- + m' + Z- - 2lm - 2mn - 2nl) 

X (c-n- + b-m- + aH- — 2ablm — 2bcmn — 2canl) 

— {{en- + bm- + at- — Im (a + b) — mn {b + e) — nl{c + a)Y 

= — 4:lmn {(a — b)(a — e)l + {b- a) (b — c) m + (c - a) (c - b) n}. 



Then 



C3^ (XP + /U.Q + (Ix + my + nz) t} = \ 



24] in Elimination hehveen Quadratic Functions. 141 

Now to obtain the resultant of 

xy + yz + zx = 0, 

ca:y + azy + hxz = 0, 

Ix + my + nz = 0, 

we need only find the four systems in their lowest terms of x:y:z, which 
satisfy the first two equations, and multiply the four linear functions obtained 
by substituting these values of x, y, z in the fourth : the product will contain 
the resultant of the system affected with some numerical factor. In the 
present case, the four systems of x, y, z are 

^• = 0, y = Q, 5=1, 

y = 0, Z=0, X = l, 

5 = 0, x = Q, y = l, 

X = {a — h) (a - c), y = (h — a) {b — c), z = {c- a) (c — 6), 

and accordingly the product of 

Ixi + my I + nzi, 

lx\ + my., + 7iZo, 

Ixs + my 3 + nzs, 

Ixi + my^ + nz^, 
becomes 

Imn {{a — b) (a — c)l + (b—a)(b — c) m + (c — a) (c — b) ?i), 

agreeing with the result obtained by my theorem, — a special numerical 
factor 4, arising from the peculiar form of the equations, having disappeared 
from the resultant. 

A geometrical demonstration may be given of the theorem which is 
instructive in itself, and will suggest a remarkable extension of it to 
functions containing more than three letters; the equation 

a^ [XU + fiV + (Ix + my + nz) t] = 0, 

which is a quadratic equation in \ : /j,, may easily be shown to imply that the 
conic \U + /jiV is touched by the straight line 

Ix + my + nz = 0. 

And we thus see that in general two conies, 

XU + fiV^O, 

passing through the intersections of two given conies, 

^7=0, F=0, 



142 



On a neiv Class of Theorems 



[24 



may be drawn to touch a given line. If, however, the given line passes 
through any of the four points of intersection, in such case only one 
conic can be drawn to touch it ; accordingly 

a a{\U -\- iJ.V + {lx + my-\-nz)t] 
must be zero when I, in, n are so taken as to satisfy this condition, that is, if 

Ix-^ + myi + nzi = 0, 
or Ix^ + viy^ + nz.2 = 0, 

•or lx« + my-i + nz^ = 0, 

■or Ixi + myi + nZi = 0, 

whence the theorem. 

Now suppose JJ and V to be each functions of four letters, x, y, z, t; 
when 

a {\U+ fjiV+{lx + my + nz+'pt)u] = 0. 

the conoid X[7'+^F touches the plane 

Ix + my + nz + pt = Q ; 
-and D = being a cubic equation, in general three such conoids can be drawn. 
Considerations of analogy make it obvious to the intuition, that in the 
particular case of two of these becoming coiacident, the given plane 

Ix 4- fiiy + nz + pt 
must be a tangent plane to those two coincident conoids at one of the points 
where it meets the intersections of U=0, 1^ = ; that is 

Ix + my + nz +pt = 
will pass through a tangent line to, or in other words, may be termed 
a tangent plane to the intersections. Hence the following analytical 
theorem, derived from supposing q, r, s, t to be proportional to the areas 
of the triangular faces of the pyramid cut out of space by the four 
•coordinate planes to which x, y, z, t refer. As these planes are left 
-indefinite, q, r, s, t are perfectly arbitrary. 

Theorem. The resultant of 
1. U=0) 



F=0 



where U and V are functions of x, y, z, t ; 

Ix + my + 7iz + pt = ; 
dU dU dU dU 



= 0; 



dx ' 


dy' 


dz' 


dt 


dV 

dx ' 


dV 
dy' 


dV 
dz' 


dV 
dt 


I, 


TO, 


n, 


P 


S> 


■'>'■, 


-s, 


t 



24] in Elimination between Quadratic Functions. 143 



which system, it will be observed, consists of three quadratic functions, and 
one linear function of x, y, z, t, contains the factor 

a C3^ {\U+ /j,V+ {Ix + my + nz + pt) u}. 

This last quantity is of the 4 x 3th, that is, the 12th order in respect of 
the coefficients in U and V combined ; of the 4 x 2th, that is, the 8th order 
in respect of I, m, n, p ; and of the zero order in respect of q, r, s, t. 

The resultant which contains it is of the (4 + 4 + 2 . 4)th, that is, 16th 
order in respect to the coefficients in U and V; of the (4 + 8)th, that is, the 
12th, in respect of I, m, n,p; and of the 4th in respect of q, r, s, t. Hence 
the special (and, as far as the geometry of the question is concerned, the 
unnecessary, I may not say extraneous or irrelevant) factor which enters .into 
the resultant is of the 4th order in respect to the combined coefficients of U 
and V* ; and of the same order in respect to I, m, n, p, and in respect to 
q, r, s, t. 

I have not yet succeeded in divining its general value. 

In the very particular example, of the system, 

axe^ + /3t/2 = 0, 

C2- + df = 0, 

Ix + my + nz + pt = 0, 

/Sy, 0, 
cz, 



0, 



0, 



dt 







0, 



I find that the double determinant is 

c-d-a.-^'' {cp- + dn°y (m-a + Z-/3)-, 
and the resultant is 

q*c-d-a.-^' (cp"- + dn-y {m'a. + P^f, 

giving as the special factor 

q"^' {cp' + dn^f. 

I believe that the theorem which I have here given for determining the 
condition that Ix -\- my + nz + pt shall be a tangent plane to the intersection 
of two conoids U and V, nam,ely, that the determinant of 

\U + jjbV -\- {Ix -\- my + nz + pt) u 

shall have two equal roots, is altogether novel. 

* And consequently of the second in respect to the separate coefficients of each. 



144 On a neiv Class of Theorems in Elimination. [24 

What is the meaning of all three roots of this determinant becoming 
equal, that is, of only one conoid being capable of being drawn through the 
intersection of U and V to touch the plane 

Ix + my + nz + pt ? 

Evidently {ex vi analogue) that this plane shall pass through three con- 
secutive j3oints of the curve of intersection, that is, that it shall be the 
osculating plane to the curve. 

If we return to the intersection of two co-planar conies, and if we suppose 
a line to be drawn through two of the points of intersection, the conies 
capable of being drawn through the four points of intersection to touch the 
line, besides becoming coincident, evidently degenerate each into a pair 
of right lines. It would seem, therefore, by analogy, that if a plane be 
drawn including any two tangent lines to the curve of intersection of two 
surfaces of the second degree, this should be touched by two coincident 
cones drawn through the curve of intersection, and consequently every such 
double tangent plane to the intersection of two conoids (and it is evident 
that one or more of these can be taken at every point of the curve) must 
pass through one of the vertices of the four cones in which the intersection 
may also be considered to lie ; and it would appear from this, that in general 
four double tangent planes admit of being drawn to the curve, which is the 
intersection of two conoids, at each point thereof At particular points 
a tangent plane may be drawn passing through more than one of the 
vertices, and then of course the number of double tangent planes that can 
be drawn will be lessened. These results, indicated by analogy, become 
immediately apparent on considering the curve in question as traced upon 
any one of the four containing cones. For the plane drawn through a 
tangent at any point, and the vertex of the cone being a tangent plane 
to the cone, must evidently touch the curve again where it meets it. We 
thus have an additional confirmation of the analogy between a point of 
intersection of two curves and the tangent at any point of the intersection 
of two surfaces. 

I might extend the analytical theorems which have been established for 
functions of three and four to functions of a greater number of variables; 
but enough has been done to point out the path to a new and interesting 
class of theorems at once in elimination and in geometry, which is all that 
I have at present leisure or the disposition to undertake. 



25. 



ADDITIONS TO THE ARTICLES*, "ON A NEW CLASS OF 
THEOREMS," AND "ON PASCAL'S THEOREM." 

[Philosophical Magazine, xxxvil. (1850), pp. 363 — 370.] 

FinST addition. — I have alluded in the second of the above articles to a 
more general theorem, comprising, as a particular case, the theorem there 
given for the simultaneous evanescence of two quadratic functions of 
2?i letters, on n linear equations becoming instituted between the letters. 

In order to make this generalization intelligible, I must premise a few 
words oij the Theory of Orders, a term which I have invented with particular 
reference to quadratic functions, although obviously admitting of a more 
extended application. A linear function of all the letters entering into a 
function or system of functions under consideration I call an order of the 
letters, or simply an order. Now it is clear that we may always consider 
a function of any number of letters as a function of as many orders as there 
are letters ; but in certain cases a function may be expressed in terms of 
a fewer number of orders than it has letters, as when the general character- 
istic function of a conic becomes that of a pair of crossing lines or a pair of 
coincident lines, in. which event it loses respectively one and two orders, and 
so for the characteristic of a conoid becoming that of a cone, a pair of planes 
or two coincident planes, in which several events, a function of four letters 
becomes that of only three orders, or two orders, or one order, respectively. 
When a function may be expressed by means of r orders less than it contains 
letters, I call it a function minus r orders. I now proceed to state my 
theorem. 

Let U and V be functions each of the same m letters, and suppose that 
the determinant in respect of those letters of U+fiV contains i pairs of 

* [* pp. 138, 139 above. Ed.] 

S. 10 



146 On a neio Class of Theorems. [25 

equal linear factors of jjl ; then it is possible, by means of i linear equations 
instituted between the letters, to make U and V each become functions of 
the same m — 2i orders ; and conversely, if by i equations between the letters 
U and V may be made functions of the same m — 2i orders, the determinant 
of f + yu.F considered as a function of /x will contain i square factors. 

Thus when m = 2n and i = n, U and V will each become functions of 
zero orders, that is, will both disappear, provided that on the institution of 
a certain system of n linear equations, among the letters of which U and V 
are functions, the determinant of {U+/j,V) is a perfect square, — which is 
the theorem given in the article referred to. 

So for example if U and V be quadratic functions of four letters, and 
therefore the characteristics of two conoids, a {U + /jlV) being a perfect square, 
expresses that these conoids have a straight line in common lying upon each 
of their surfaces. 

If U and V be quadratic functions of three letters only, and admit there- 
fore of being considered as the characteristics of two conies, a {U + fj,V) 
containing a square factor, is indicative of these conies having a common 
tangent at a common point, that is, of their touching each other at some 
point ; for it is easily shown that the disappearance of two orders from any 
quadratic function by virtue of one linear function of its letters being zero, 
indicates that the line, plane, &c. of which the linear function is the 
characteristic is a tangent to the curve, surface, &c. of which the quadratic 
function is the characteristic. 

I pass now to a generalization of the theorem which shows how to 
express, under the form of a double determinant, the resultant of one linear 
and two quadratic homogeneous functions of three letters (which I should 
have given in the original paper, had I not there been more intent upon 
developing an ascending scale than of expatiating upon a superficial ramifi- 
cation of analogies), and which constitutes my Second addition to that paper, 
to wit — 

If U and V be homogeneous quadratic, and ij, X, ... L^ homogeneous 
linear functions of (?i + 2) letters oa^, x^ ... 00^+2, the determinant of the entire 
system of m -f 2 functions is equal to 

□ I \ [xU + ij.V+Liti + L«t2 + ...+Lntn}; 

the demonstration is precisely similar to the analytical one given in the 
September Number* for the particular case of m = l. 

When w = 0, we revert to Mr Boole's theorem of elimination between U 
and V already adverted to. The proof, it will be easily recognized, does not 
require the application of the more general theorem relative to the simul- 
[* p. 140 above.] 



25] On a new Class of Theorems. 147 

taneous depression of orders of two quadratic functions, but only the limited 
one before given, which supplies the conditions of their simultaneous 
disparition. I now proceed to develope more particularly certain analogies 
between the theory of the mutual contacts of two conies, and that of the 
tangencies to the intersection of two conoids. 

But here again I must anticipate some of the results which will be given 
in my forthcoming memoir on Determinants and Quadratic Functions, 
by explaining what is to be understood by minor determinants, and the 
relation in which they stand to the complete determinant in which they are 
included. This preliminary explanation, and the statement of the analogies 
above alluded to, will constitute my Third and last addition. 

Imagine any determinant set out under the form of a square array of 
terms. This square may be considered as divisible into lines and columns. 
Now conceive any one line and any one column to be struck out, we get 
in this way a square, one term less in breadth and depth than the original 
square ; and by varying in every possible manner the selection of the line 
and column excluded, we obtain, supposing the original square to consist of 
n lines and n columns, ii' such minor squares, each of which will represent 
what I term a First Minor Determinant relative to the principal or complete 
determinant. Now suppose two lines and two columns struck out from the 

original square, we shall obtain a system of \ -^-^ A squares, each two 

terms lower than the principal square, and representing a determinant of 
one lower order than those above referred to. These constitute what I term 
a system of Second Minor Determinants; and so in general we can form 
a system of rth minor determinants by the exclusion of r lines and r 
columns, and such system in general will contain 

[ n {n-l) ...{n-r+l) y 
\ 1.2 ...r j 

distinct determinants. 

I say " in general " ; because if the principal determinant be totally or 
partially symmetrical in respect to either or each of its diagonals, the 
number of distinct determinants appertaining to each system of minors will 
undergo a material diminution, which is easily calculable. 

Now I have established the following law : — 

The whole of a system of rth minors being zero, implies only {r + 1)" 
equations, that is, by making (r + 1)^ of these minors zero, all will become 
zero ; and this is true, no matter what may be the dimensions or form of the 
complete determinant. But furthermore, if the complete determinant be 
formed from a quadratic function, so as to be symmetrical about one of its 
diagonals, then \ (r + 1) (r + 2) only of the rth minors being zero, will serve 

10—2 



148 On a new Cla^s of Theorems. [25 

to imply that all these minors aite zero. Of course, in applying these 
theorems, care must be taken thatj the (r + iy or |(r + l)(r + 2) selected 
equations must be mutually non-implicative, and shall constitute indepen- 
dent conditions. 

In the application I am about to make of these principles, we shall have 
only to deal with a system of fij'st minors and of a symmetrical determinant. 
If three of these properly selected be zero, from the foregoing it appears 
that all must be zero. 

Now let U and V be characteristics of two conies, that is, let each be 
a function of only three letters, it may be shown (see my paper* in the 
Cambridge and Dublin Mathematical Journal for November, 1850) that the 
different species of contacts between these two conies will correspond to 
peculiar properties of the compound characteristic U + jjuV. 

If the determinant of this function have tw^o equal roots, the conies 
simply touch ; if it have three equal roots, the conies have a single contact 
of a higher order, that is, the same curvature ; if its six first minors 
become zero simultaneously for the same value of //., the conies have a double 
contact. If the same value of fj,, which makes all these first minors zero, 
be at the same time not merely a double root (as of analytical necessity 
it always must be) but a treble root of 

n(t^+^F) = 0, 

then the conies have a single contact of the highest possible order short of 
absolute coincidence, that is, they meet in four consecutive points. 

The parallelism between this theory and that of two quadratic functions 
P, Q, and one linear function if of four letters, say x, y, z, t, is exact |. 
For let P + Lu + fiQ be now taken as our compound characteristic (a func- 
tion, it will be observed, of five letters, x, y, z, t, u); if its determinant have 
two equal roots, L has two consecutive points in common with the inter- 
section of P and Q, that is, passes through a tangent to that intersection ; 
if it have three equal roots, L has three consecutive points in common with 
the said intersection, that is, is an osculating plane thereto ; if its fifteen 
first minors admit of all being made simultaneously zero, L has a double 
contact with the intersection of P and Q, that is, it is a tangent plane to 
some one of the four cones' of the second order containing this intersection ; 

[* p. 119 above.] 

t Observe that P=0, § = 0, I/ = now express the equations to two conoids and a plane 
respectively. 

X This paraUelism may be easily shown analytically to imply, and be implied, in tlie 
geometrical fact, that the contact of the plane L with the intersection of the two surfaces P and 
Q, is of exactly the same kind as the contact (which must exist) between the two conies which 
are the intersections of P and Q respectively with the plane L. 



26] On a new Class of Theorems. 149 

if the same linear function of fi which enters into all these first minors be 
contained cubically in the complete determinant, then the plane L passes 
through four consecutive points of the intersection of P and Q, and the 
points where it meets the curve will be points of contrary plane flexure ; 
and, as it seems to me, at such points the tangential direction of the curve 
must point to the summit of one or other of the four cones above 
alluded to*. In assigning the conditions for L being a double tangent 
plane to the intersection of P and Q, we may take any three independent 
minors at pleasure equal to zero. One of these may be selected so as to be 
clear of the coefficients of Z ; in fact, the determinant of P + fjuQ will be 
a first minor of P + /Jt-Q + Lii ; fi may thus be determined by a biquadratic 
equation ; and then, by properly selecting the two other minors, we may 
obtain two equations in which only the first powers of the coefficients of 
X, y, z,tmL appear, and may consequently obtain L under the form of 

{ae + a) « + {he + yS) y + (ce + 7) ^ + {de + h) t, 

where a, a; 6, /3 ; c, 7 ; d,h will be known functions of any one of the four 
values oi fi. The point of contact being given will then serve to determine 
e, and we shall thus have the equation to each of the four double tangent 
planes at any given point fully determined. 

In the foregoing discussions I have freely employed the word character- 
istic without previously defining its meaning, trusting to that being apparent 
from the mode of its use. It is a term of exceeding value for its significance 
and brevity. The characteristic of a geometrical figuref is the function 
which, equated to zero, constitutes the equation to such figure. Plticker, 
I think, somewhere calls it the line or surface function, as the case may 
be. Geometry, analytically considered, resolves itself into a system of rules 
for the construction and interpretation of characteristics. One more remark, 
and I have done. A very comprehensive theorem has been given at the 
commencement of this commentary, for interpreting the effect of a complete 
determinant of a linear function of two quadratic functions {U + i^V), having 

* If this be so, then we have the following geometrical theorem : — " The summit of one of the 
four cones of the second degree xoliich contain the intersections of two surfaces of the second order 
drawn in any manner respectively through tioo given conies lying in the same plane, and having 
with one another a contact of the third degree, U'ill always he found in the same right line, namely 
in the tangent line to the two given conies at the point of contact." 

+ More generally, the characteristic of any fact or existence is the function which, equated to 
zero, expresses the condition of the actuality of such fact or existence. 

Perhaps the most important pervading principle of modern analysis, but which has never 
hitherto been articulately expressed, is tliat, according to which we infer, that when one fact of 
whatever kind is implied in another, the characteristic of the first must contain as a factor the 
characteristic of the second ; and that when two facts are mutually involved, their characteristics 
will be powers of the same integral function. 

The doctrine of characteristics, applied to dependent systems of facts, admits of a wide 
development, logical and analytical. 



150 071 a new Class of Theorems. [25 

one or more pairs of equal factors (e + e^). But here a far wider theory 
presents itself, of which the aim should be to determine the effect and 
meaning of this determinant, having any amount and distribution of multi- 
plicity whatsoever among its roots. Nor must our investigations end at 
that point; but we must be able to determine the meaning and effect of 
common factors, one or more entering into the successive systems of viinor 
determinants derived from the complete determinant of U + fiV. 

Nor are we necessarily confined to two, but may take several quadratic 
functions simultaneoi,isly into account. 

Aspiring to thes^ wide generalizations, the analysis of quadratic functions 
soars to a pitch from whence it may look proudly down on the feeble and 
vain attempts of geometry proper to rise to its level or to emulate it in its 
flights. 

The law which I have stated for assigning the number of independent, 
or to speak more accurately, non-coevanescent determinants belonging to 
a given system of minors, I call the Homaloidal law, because it is a corollary 
to a proposition which represents analytically the indefinite extension of 
a property common to lines and surfaces to all loci (whether in ordinary 
or transcendental space) of the first order, all of which loci may, by an 
abstraction derived from the idea of levelness common to straight lines and 
planes, be called Homaloids. The property in question is, that neither two 
straight lines nor two planes can have a common segment ; in other words, 
if ?! independent relations of rectilinearity or of coplauarity, as the case 
may be, exist between triadic groups of a series of w + 2, or between tetradic 
groups of a series of n + S points respectively, then every triad or tetrad 
of the series, according to the respective suppositions made, will be in 
rectilinear or in plane order. So, too, if ?i independent relations of coincidence 
exist between the duads formed out of w + 1 points, every duad will con- 
stitute a coincidence. 

This homaloidal law has not been stated in the above commentary in 
its form of greatest generality. For this purpose we must commence, not 
with a square, but with an oblong arrangement of terms consisting, suppose, 
of m lines and n columns. This will not in itself represent a determinant, 
but is, as it were, a Matrix out of which we may form various systems of 
determinants by fixing upon a number p, and selecting at will p lines and p 
columns, the squares corresponding to which may be termed determinants 
of the ^th order. We have, then, the following proposition. The number of 
uncoevanescent determinants constituting a system of the pth order derived 
from a given matrix, n terms broad and 7n terms deep, may equal, but can 
never exceed the number 

(n-p + l){m—p + l). 



25] On Pascal's Theorem. 151 



Remark on Pascal's and Brianchon's Theorems. 

I omitted to state, in the September Number of the Journal *, that the 
demonstration there given by me for Pascal's, applied equally to Brianchon's 
theorem. This remark is of the more importance, because the fault of the 
analytical demonstrations hitherto given of these theorems has been, that 
they make Brianchon's consequence of Pascal's, instead of causing the two 
to flow simultaneously from the application of the same principles. No 
demonstration can be held valid in method, or as touching the essence of the 
subject-matter, in which the indifference of the duadic law is departed from. 
Until these recent times, the analytic method of geometry, as given by 
Descartes, had been suffered to go on halting as it were on one foot. To 
Pliicker was reserved the honour of setting it firmly on its two equal 
supports by supplying the complementary sj'stem of coordinates. This 
invention, however, had become inevitable, after the profound views pro- 
mulgated by Steiner, in the introduction to his Geometry, had once taken 
hold of the minds of mathematicians. To make the demonstration in the' 
article referred to apply, totidem Uteris, to Brianchon's theorem (recourse 
being had to the correlative system of coordinates), it is only needful to 
consider U as the characteristic of the tangential envelope of the conic, 
X, y, z, t, u, V as the characteristics of the six points of the circumscribed 
hexagon, j> the characteristic of the point in which the line x, v meets the 
line 2, t; ay — au will then be shown to characterize the point in which 
t, as meets v, z \ and thus we see that y, u\ t, x; v, z, the three pairs of 
opposite sides of the hexagon, will meet in one and the same point, which 
is Brianchon's theorem. 

[* p. 138 above.] 



26. 



ON THE SOLUTION OF A SYSTEM OF EQUATIONS IN WHICH 
THREE HOMOGENEOUS QUADRATIC FUNCTIONS OF THREE 
UNKNOWN QUANTITIES ARE RESPECTIVELY EQUATED 
TO NUMERICAL MULTIPLES OF A FOURTH NON-HOMO- 
GENEOUS FUNCTION OF THE SAME. 

[Philosophical Magazine, xxxvii. (1850), pp. 370 — 373.] 

Let U, V, W be three homogeneous quadratic functions of x, y, z, and 
let ftj be any function of x, y, z of the ?ith degree, and suppose that there 
is given for solution the system of equations 

W = Ceo. 

Theorem. The above system can be solved by the solution of a cubic 
equation, and an equation of the ?ith degree. 

For let D be the determinant in respect to x, y, z of 
fV+gV+hW, 
then Z) is a cubic function of/, g, h. Now make 

D=0, Af+Bg+Ch = 0; 
the ratios o( f:g :h which satisfy the last two equations can be determined 
by the solution of a cubic equation, and there will accordingly be three 
systems of f, g, h which satisfy the same, as 

/i, 9i, k, 
/., g., h, 
fs, g^, K 

Now Z) = implies that/f7 +gV+ hW breaks up into two linear factors; 
accordingly we shall find 

(liic + 7?i,y -|- TiiS) {\iX + fi^y -(- Viz) = 0, 
(l^x -f- m^y + n^z) {X^oo + n^y + v^z) = 0, 
(4« 4- m-^y + ihz) {\,x + ix^y + v^z) = 0, 



26] On the solution of a System of Equations. 153^ 

in which the several sets of I, m, n ; \, fi, v can be expressed without diffi- 
culty in terras of the several values of \Jf, \/g, \Jh. 
Let the above equations be written under the form 
PP' = 0, 

QQ' = 0, 

RR' = Q. 

Since the given equations are perfectly general, it is readily seen that 
the equations 

(P = 0, P' = 0), {Q = 0,Q' = 0), {R = 0,R' = 0), 
will severally represent pairs of opposite sides of a quadrangle expressed by 
general coordinates x, y, z; so that one of the two functions R, R' will be a- 
linear function of P and Q and also of P' and Q', and the other will be a 
linear function of P and Q' and also of P' and Q*. 

In order to solve the equations, we need only consider two such pairs- 
as PP' = 0, QQ' = ; we then make 

P = 0, Q = 0, 
or P = 0, Q' = 0, 

or P' = 0, Q=0, 

or P' = 0, Q' = 0. 

Any one of these four systems will give the ratios of x:y:z; and then,, 
by substitution in any one of the given equations, we obtain the values of 
X, y, z by the solution of an ordinary equation of the ?ith degree. The 
number of systems x, y, z is therefore always 4)i. 

The equations connected with the solution of Malfatti's celebrated 
problem, " In a given triangle to inscribe three circles such that each circle 
touches the remaining two circles and also two sides of the triangle," given 
by Mr Cayley in the November Number for 1849 of the Cambridge and 
Dublin Mathematical Journal, to wit, 

by"- + cz"- + 'i.fyz = Q-a (be -/-) = A, 

cz" + ax'' + 2gzx = 6-b (ca — g^) = B, 

ax' + by'' + Ihxy = d'c (ab - h^) = G, 
come under the general form which has just been solved. It so happens, 
however, that in this particular case 



f> 


9i> 


K 


/=. 


92, 


lu 


u 


g,. 


h 



* Were it Bot for this being the ease, the number of solutions would be n times the number 
of ways of obtaining duads out of three sets of two things, excluding the duads forming the sets, 
that is, the number of solutions would be Xin in place of in, the true number. 



154 071 the sohction of a St/stem of Equations. [26 

become respectively 



\ 


1 


1\ 


), 


B' 


C 


1 




1 




0, 




B' 




c 


1 


1 











C' 


B' 





and the cubic equation is resolved without extraction of roots. 

It follovs^s from my theorem that the eight intersections of three con- 
centric surfaces of the second order can be found by the solution of one cubic 
and one quadratic equation ; and in general, if we have (fi, ^jr, any three 
quadratic functions of x, y, z, and (f) = 0, -\|r = 0, ^ = be the system of 
equations to be solved, provided that we can by linear transformations 
express (j>, -i|r, 6 under the form of 

U — aiu, 

V-hw, 

W — ciu, 

U, V, W being homogeneous functions, and lu a non-homogeneous function 
of three new variables, x', y', z , we can find the eight points of intersection 
of the three surfaces, of which TJ, V, W are the characteristics, by the 
solution of one cubic and one quadratic. But (as I am indebted to Mr Cayley 
for remarking to me) that this may be possible, implies the coincidence 
of the vertices of one cone of each of the systems of four cones in which the 
intersections of the three surfaces taken two and two are contained. 

I may perhaps enter further hereafter into the discussion of this elegant 
little theory. At present I shall only remark, that a somewhat analogous 
mode of solution is applicable to two equations, 

U=aP% 

V=bP-, 

in which U, V are homogeneous quadratic functions, and P some non-homo- 
geneous function of x, y. 

We have only to make the determinant of /fZ-f-^'F equal to zero, and we 
shall obtain two systems of values ofy, g, wherefrom we derive 

l^x+ nhy = ± ^/{afi + hg^) P, 

l^x + m„y = + V(«/2 + %) P, 

from which x and y may be determined. 



27. 



ON A PORISMATIC PROPERTY OF TWO CONICS HAVING 
WITH ONE ANOTHER A CONTACT OF THE THIRD ORDER. 

[Philosophical Magazine, XXXVII. (1850), pp. 438, 439.] 

If two conies have with one another a contact of the third order, that is, 
if they intersect in four consecutive points, it will easily be seen that their 
characteristics referred to coordinate axes in the plane containing them must 
be of the relative forms ocr + yz, k{y- + x- + yz) respectively, y characterizing 
their common tangent at the point of contact*. 

Hence if we take planes of reference in space, and call t the characteristic 
of the plane of the conies, the equations to any two conoids drawn through 
them respectively will be of the relative forms 

JJ = X- + yz + tu = 0, 

- V=y- + x'^ + y2 + tv = 0. 

Using W to denote V— IT, and (W) to denote what W becomes when ey 
is substituted for t, we see that W and (Tf) are of the respective forms 
y- + tw and yO ; showing that the former is the characteristic of a cone which 
will be cut by any plane t— ey drawn through the line {t, y) in a pair 
of right lines ; or, in other words, that one of the cones containing the inter- 
section of the two variable conoids ( V and U) will have its vertex in the 
ijivariable line which is the common tangent to the two fixed conies : this 
proves the theorem stated by me hypothetically in a foot-note in one of my 
papers in the last number of the Magazine-^. The steps of the geometrical 
proof there hinted at are as follows. 

* These relative or conjugate forms are taken from a table whicli I shall publish in a future 
number of this Magazine, exhibiting the conjugate characteristics in their simplest forms, 
correspondent to all the various species of contacts possible between lines and surfaces of the 
second degree. This table is as important to the geometer as the fundamental trigonometrical 
formulte to the analyst, or the multiplication table to the arithmetician ; and it is surprising 
that no one has hitherto thought of constructing such. 

[t p. 149 above. ] 



156 On a Porismatic Property of two Conies. [27 

The four consecutive points in which the two conies intersect will be 
consecutive points in the curve of intersection of the two variable conoids. 
This curve lies in each of four cones of the second degree. Every double 
tangent plane to it passes through the vertex of one amongst these. The 
plane containing four, that is, two (consecutive) pairs of consecutive points, 
is a double tangent plane, and will therefore pass through a vertex ; but four 
consecutive points of a curve of the fourth order described upon a cone, 
and lying in one tangent plane thereto, can only be conceived generally 
as disposed in the form of an /, of which the belly part will point to the 
vertex ; or, in other words, at any point where two consecutive osculating 
planes coincide so that the spherical curvature vanishes, the linear curvature 
will also vanish, that is, there will be a point of inflexion at which, of course, 
the tangent line must pass through the vertex of the cone. This is the 
assumption felt to be true, but stated by me hypothetically in the paper 
referred to, because a ready demonstration did not at the moment occur 
to me. The legitimacy of this inference is now vindicated by the above 
analytical demonstration. 

The methods of general and correlative coordinates and of determinants 
combined possess a perfectly irresistible force (to which I can only compare 
that of the steam-hammer in the physical world) for bringing under the 
grasp of intuitive perception the most complicated and refractory forms of 
geometrical truth. 



28. 



ON THE ROTATION OF A RIGID BODY ABOUT A 
FIXED POINT. 

[Philosophical Magazine, xxxvii. (1850), pp. 440 — 444.] 

In the Cambridge and Dublin Mathematical Journal for March 1848, 
an article by Professor Stokes, of the University of Cambridge, is ushered in 
"with the words following : — 

"The most general instantaneous* motion of a rigid body moveable in 
all directions about a fixed point consists in a motion of rotation about an 
axis passing through that point. This elementary proposition is sometimes 
assumed as self-evident, and sometimes deduced as the result of an analytical 
process. It ought hardly perhaps to be assumed, but it does not seem 
desirable to refer to a long algebraical process for the demonstration of 
a theorem so simple. Yet I am not aware of a geometrical proof anywhere 
published which might be referred to." 

The learned and ingenious professor is indubitably right, and might have 
trusted himself to assert less hesitatingly the necessity of demonstrating 
this proposition, which possesses none of the characters of a self-evident 
truth ; but it is to be regretted that he should have stated it in such a form 
as naturally to lead the incautious reader to mistake the nature and grounds 
of its existence, which consist in this fact — that any kind of displacement 
of a body moveable about a fixed axis, whether instantaneous and infini- 
tesimal, or secular and finite, is capable of being effected by a single rotation 
about a single axis. 

The annexed simple proof of this capital law has the advantage of afford- 
ing a rule for compounding into one any two (and therefore any number of) 
rotations given in direction, magnitude and order of succession. 

* The italics do not exist in the original. 



158 On the Rotation of a [28 

It will somewhat conduce to simplicity if we fix our attention upon a 
spherical surface rigidly connected with the rotating body, and having its 
centre at the fixed point thereof. When the positions of two points in 
this are given, the position of the body is completely determined. 

Now evidently two points A, B may be brought respectively to A'B' 
(if AB = A'B') by two rotations ; the first taking place about a pole situated 
anywhere in the great circle bisecting AA' at right angles, the second about 
A', the position into which it is brought by the first rotation. This view 
leads us to consider the effect of two rotations taking place successively 
about two axes fixed in the rotating body. Or again, we may make the plane 
A'B' revolve into the position AB round a pole taken at the node in which 
the two planes intersect, and then the points A, B swing into their new 
positions A', B' hj means of a rotation about the pole of the great circle, 
of which A'B' forms a part. This mode of effecting the displacement 
naturally suggests the consideration of the effect of rotations taking place 
successively about two axes fixed in space. 

First, then, let us study the effect of the combination of a rotation 
(a) having P for its pole, followed by another (/3), of which Q is the pole, 
P and Q being points in the surface of the revolving sphere. 

In drawing the annexed figure, I have supposed that the two rotations 
are of the same kind, each tending, when a 
spectator is standing with his head to the 
respective poles and his feet to the centre, to 
make a point at his right-hand pass in front of 
his face towards his left-hand. Let now PQ 

revolve through ^ positively into the position of 


PR, and through ^ negatively into that of QR. 

Then I say that the two impressed rotations a 
and /8 about P and Q will be equivalent to a single rotation about R, equal 
to twice the acute angle between QR, RP. 

Let the first rotation about P bring Q to Q' and R to R' ; it is clear that 
QPR, Q'PR, Q'PR' are all equal triangles. Therefore R'Q'R = 2PQR = /3. 
Consequently the positive rotation /3 about Q' (the new position of Q) will 
carry R' back again to R, its original position. Hence the actual motion 
which results from the successive rotations combined being consistent with 
jR remaining at rest, must be equivalent to a single rotation about R. 

To find its magnitude, let the second rotation carry P to P'*; then the 
angular displacement PRP' (which is the required rotation of the whole 

* The reader is requested to fill in the point P' and join P'E. 




28] Rigid Body ahout a Fixed Point. 159 

body) is equal to twice the acute angle between Q'i2, RF, which is the same 
as that between QIL, RP, as was to be shown. Thus we see that the semi- 
rotations about three poles (considered as the angular points of a spherical 
triangle), which, taken in order, would bring the sphere back to its first 
undisturbed position, are equal to the included angles at such poles respec- 
tively. 

If in our figure the order of the rotations had been reversed, FQr, QPr 
would have been taken respectively equal to PQR, QPR, but on the opposite 
side of PQ, and r would have been the resultant pole, the resultant rotation 
remaining in amount the same as before. 

If either of the rotations had been negative, the resultant pole would be 
found in QR produced, namely, at the intersection of rQ or rP with PQ. 

Calling the resultant rotation 7, we have always 

sin - : sin ^ : sin ^ :: sin QR : sin RP : sinPQ. 

When the component rotations are infinitesimal in amount, R and ?• will 
come together in QP ; the order of succession of the rotations will be 
indifferent, and we shall have 

a : /3 : 7 :: sin j^ : sin ^ : sin | :: sin QR : sin RP : sin PQ, 

which gives the rule for the parallelogrammatic composition of two simul- 
taneously impressed rotations*. 

If, next, we consider the effect of rotations about two poles, P and Q, 
fixed in space (supposing, as above, that they take place first about P and 
then about Q), we must take QPr equal to half the contrary of the rotation 
about P, and PQr to half the direct rotation about Q (the angle being now 
taken positive which was on the first supposition negative, and vice versa) ; 
so that, retaining the original figure, the first rotation will bring r to R, 
and the second carry R back to r; showing that r is the resultant pole, 
and thatf P'rP, the resultant rotation, will be double the acute angle 
between Qr, rP, as in the former case. 

To popular apprehension the important doctrine of uniaxial rotation 
may be made intelligible by the following mode of statement. Take a 
pocket-globe, open the case and roll about the sphere within it in any 
manner whatever ; then closing the case, there will unavoidably remain two 
points on the terrestrial surface touching the same two points on the celestial 
surface as they were in apposition with before the sphere was so turned about 
in its case. 

* Compare Mr Airy's Tracts, Art. " On Precession and Nutation." 
t P' is not expressed in the Ingure given. 



160 On the Rotation of a [28 

It is right to bear in mind that the whole of this doctrine is comprised 
4n, and convertible with, the following easy geometrical proposition relative 
to arcs of great circles on any spherical surface, including the plane as an 
'extreme case. 

" The arcs joining the extremities (each with each in either order) of two 
other equal arcs, subtend equal angles at either of the points of intersection 
■of two great circles bisecting at right angles the first-named connecting 
.arcs*." 

The spherico-triangular mode of compounding rotations given in the 
above simple disquisition may easily be made the parent of a whole brood of 
geometrical consequences, which, however, I must leave to the ingenuity 
and care of those who have a turn for this kind of invention. 

But I ought not to omit to invite attention to a remarkable form, which 
.may be imparted to the theorems above stated for the composition of finite 
.rotations, or rather to a theorem which may be derived from them by an 
-obvious process of inference. 

Let P, Q, R ... X, Z be any number of points on a sphere capable of 
^moving about its centre, joined together by arcs of great circles so as to form 
a spherical polygon. Imagine any number of rotations to take place about 
these points in succession as poles. It matters not which is considered the 
first pole of rotation, but the order of the circulation must be supposed given, 
as, for instance, PQR...XZ, or QR...XZP, or R ... XZPQ, &c. This 
will be one order; the reverse order -svould be PZX ...RQ, or QPZX ...R,k,c. 

I shall suppose the circulation to be of the kind first above written. 
Now we may make two 'hypotheses : — 

1. That .the poles are fixed in space. 

2. That they are fixed in the rotating body. 

In the first case, let .the rotations about the given poles P, Q,R,S ... X, Z 
"be double the amounts which would serve to tran.sport PQ to QR, QR to 
RS ... XZ to ZP respectively. 

In the second case, let the rotations be double the amounts which w'ould 
• carry PZ to ZX . . . SR to RQ, RQ to QP respectively. Then, on either 
supposition, the sum of the combined rotations is zero; or, to use a more 
convenient and suggestive form of expression, if the poles of rotation form a 
closed spherical polygon .whose angles are respectively equal to the semi- 
rotations about the poles, the resultant rotation is zero. 

* This proposition will be seen to be inamediately demonstrable, by the comparison of equal 
triangles, when viewed as the converse of this other. " The arcs (or right lines) joining the 
.correspondent extremities of the bases of two similar isosceles spherical (or plane) triangles 
-having a common vertex, are equal to each other." 



28] Rigid Body about a Fixed Point. 161 

This proposition is immediately derivable from the fundamental one 
relative to three poles, given above, by dividing the polygon into triangles 
by arcs, joining any one of the poles with all the rest, or (as pointed out to 
me by my eminent friend Prof. W. Thomson) it becomes apparent as a 
particular case of a more general proposition, on representing the motion 
about the successive axes as effected by two equal pyramids having a 
common vertex at the centre of motion, of which the one is fixed in 
space, and the other is fixed in the revolving body and rolls over the 
first, so that the corresponding equal faces are successively brought into 
coincident apposition. 

P.S. To find the pole of rotation whereby PQ may be brought into the 
position P'Q', we may use the following simple construction. 

Measure off from the node of the great circles (or right lines) con- 
taining PQ and P'Q', two distances in the proper direction upon each (four 
distinct assumptions may be made), say OR and OS equal to one another 
and to the difference between OP and OP', then the pole of rotation required, 
say E, is the centre of the circle described about ROS, and the amount of 
rotation is the angle subtended by OR or OS at E. The writer of this paper 
suggests that axis of displacement would be a convenient term for designating 
the line whereby any finite change in the position of a body moveable about 
a fixed centre may be brought about ; a geometrical theory of rotation leading 
to the investigation of a very curious species of correlation, now opens upon 
the view, the general object of which may be stated as follows : 

" Given upon a sphere or plane any curve considered as the locus of 
successive poles of instantaneous rotation, and the ratio of the rotation 
about each pole to its distance from the one that follows*, to construct 
the curve of the poles of displacement, and to determine the amount of 
rotation corresponding to each such pole." 

The discussion of this question offers a fine field for the exercise of 
geometrical taste and skill. 

* Which by analogy may be termed the " density of rotation." 



11 



29. 

ON THE INTERSECTIONS OF TWO CONICS. 

[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 18—20.] 

Let the two conies be written 

U = aaf+ by-+ cz^ + 2a'yz + Wzx + 2c'ocy = 0, 

F = ou5= + /3^= + r^z-" + layz + "i^'zx + l^xy = 0, 
and make 

U ->r\V = Aa? ^ By"- -V Cz"- + 2A'yz + Wzx + 20' xy. 
In my paper in the last number of the Journal*, I showed that the case of 
intersection of the two conies in two points was distinguishable from all other 
eases by the equation d(C7 + \F) = having two imaginary roots. When 
all the roots are real, the curves either intersect in four points or not at all. 
On the former supposition, 

-C'^ + AB, -A" + BG, -B'"- + CA, 
which are quadratic functions of X, will be negative for all three values of \. 
On the contrary supposition, one value of X will make all these three 
quantities negative, but the other two values with each make them all 
three positive. 

Hence we obtain a symmetrical criterion (which I strangely omitted to 
state in my former paper) by forming the quantity 

A'' + B" + C"-AB-AC-BC. 
A cubic equation 

Ly'+My'' + Ny + P = 
may be then constructed, of which the three values of the above function 
corresponding to three values of X will be the roots. 

The condition for real intersection is that L, M, N, P should be all of the 
same sign. The conies being supposed real, L and P are necessarily in both 
cases of the same sign. The condition is therefore satisfied if either L, M, 

[* p. 119 above.] 



29] On the Intersections of Two Conies. 163 

N, or M, N, P be of the same sign, and is consequently equivalent to the 

M N . . N M 

condition that -y and y- shall be both positive, or -^ and -^ both positive. 

It does not appear to be possible in the nature of the question to find a 
criterion for distinguishing between the two cases, dependent on the sign 
of one single function of the coefficients. 

The case of double contact, abstraction being made of binary intersection, 
is a sort of intermediary state between intersection in four points and non- 
intersection ; and accordingly, as shown in my former paper for this case, the 
two equal values of A, will make the three quantities 

AB-G'\ BG-A'\ GA-B' 
all real ; so that two of the values of y corresponding to the equal values of X 
are zero, and the criterion becomes nugatory as it ought to do. 

Again, when the two conies do not intersect, I distinguished two cases 
according as they lie each without, or one within the other, that is, according 
as they have four common tangents or none. 

But, as Mr Cay ley has well remarked to me, a similar distinction exists 
when the conies intersect in four points ; in that case also they may have 
four common tangents or not any : when they intersect in two points they 
have necessarily two and only two common tangents. There is no difficulty 
in separating these four cases. 

Let the conies be written 

(r) = A^' + Bri'-G^\ 
(U) and (F) being what U and V become when the coordinates are changed 
from X, y, z to ^, i], f. 

A, B, G are the three values of X in the equation 

n(V-XU) = 0. 
If the curves intersect A — G, B — G must have different signs, that is, G 
must be an intermediary quantity between A and B. 

Again, the tangential equations to the conies expressed by the correlative 
system of coordinates will be 

A^ B G ' 
and that these may have four real systems of roots, 

l_\ 1_1 

A G' G B 
must have the same sign ; and consequently, as ^ — C and G — B are 

11—2 



164 On the Intersections of Two Conies. [29 

supposed to have the same siga, A and B, and therefore all three A, B, G, 
have the same sign. We have therefore the following rule : 

Let the equation in X, namely, a (U + XV) = 0, he called ^ = 0, and the 
equation in y, above given, m = 0. By an equation being congruent or 
incongruent, understand that its roots have all the same sign or not all 
the same sign. 

Then to congruent, 6 congruent, implies that the intersections and 
common tangents are both real ; a> congruent, d incongruent, implies that 
the intersections are real, but the common tangents imaginary ; a> incon- 
gruent, congruent, implies that the intersections and common tangents 
are both imaginary ; « incongruent, 6 incongruent, implies that the inter- 
sections are imaginary, but the common tangents real. 

In like manner, as the cases of contact of lines are limiting cases to those 
which relate to the relative configurations of their points of intersection, so 
the cases of contact of surfaces are limiting cases in which the characters 
which usually separate the different forms of their curve of intersection exist 
blended and indistinguishable. The first step therefore to the study of the 
particular species of the curve of the fourth degree*, in which two surfaces 
of the second degree intersect, is to obtain the analytical and geometrical 
characters of their various species of contact. Accordingly I have made an 
enumeration of these different species, no less than 12 in number, many of 
them highly curious and I believe unsuspected, which the reader may consult 
in the Philosophical Magazine for February, 185 l"f. 

By the aid of these landmarks, I have little doubt, should time and leisure 
permit, of mapping out a natural arrangement of the principal distinctions of 
form between that class at least of lines in space of the fourth order which 
admit of being considered the complete intersection of two surfaces. 

* I have found that the 16 points of spherical flexure in this curve are the four sets of four 
points in which it meets the four faces of the pyramid whose summits are the vertices of the four 
cones of the second degree in which the curve is completely contained, which 16 points reduce to 
4 when the two surfaces have an ordinary contact, and to 1 when they have a cuspidal contact : 
of course in the ease of contact the pyramid above described in a manner folds up aud vanishes, 
as there are no longer i distinct vertices. I have found also that when the factors of D (U+\V)y 
{U and V being the characteristics of the two surfaces) are all unreal, the points of flexure are all 
unreal. When two factors are real and two imaginary, two of the faces of the pyramid (namely, 
its two real faces) wiU each contain one {and only one) pair of real points of flexure, and the other 
two planes none; and lastly, when the factors of D {U+\V) are all real, then either all the points 
of flexure are imaginary, or else all the eight contained in a certain two of the pyramidal faces 
are real : and these two cases admit of being distinguished by a method analogous in its general 
features to that whereby I have shown in the text above how to distinguish between the cases of 
4 real and 4 imaginary points of intersection of two conies. Where the two surfaces have an 
ordinary contact, the curve of intersection, it is well known, has a double point; and where the 
surfaces have a higher contact, the curve has a cusp. Thus in the fact of the 16 flexures reducing 
to 4 and to 1 in these respective cases, we see a beautiful analogy to what takes place with the 9 
flexures of a plane curve of the third degree, which contract to 3 and 1, according as the curve 
has a double point or a cusp. 

[t p. 219 below.] 



30. 



ON CERTAIN GENERAL PROPERTIES OF HOMOGENEOUS 
FUNCTIONS. 

[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 1 — 17.] 

Let ;^ denote the operation 



and A the operation 



d d d 

X-i -^ + Xn -^ r ■ . . ~r X-Yi ~^ , 

da-y da^ dan 



'''^i + ''^5^+---^''"^- 



and now suppose that w, a homogeneous function of i dimensions of 
Oi, a«...an, and not of any of the quantities x^, x^... Xn, is subjected to 
the successive operations indicated by A^x''. 

We have 

A^y^o) = A^~'^Ax''a>, 

A .. ( d d d \ ( d d d \'' 

/ d d d \ ^ , 

= r(c — r+ l)')(J'~^ci), 

for x^~^'^ is of (r — 1) dimensions, lower than oo (which is of i dimensions) in 
tti, a.2... a„. 

Hence 

A^X'^ = r (t - r + 1 ) Jl«-1;;^'-i<b 

= &c. = {r(r-l)...(r-s+l)} 

{(.-r + l)(.-7- + 2)...(.-r + s)l;^:'--a.. (1) 



166 On certain General Properties [30 

Now in the expression 

suppose that we write 

ajj =«j + tij6, 

a^a = M2 + «2e, 

we have, by Taylor's theorem, 

X''q) = U^'ft) + A TJ-'m € + A^lfw -— 2 + ... + A'-Wo} „ , 

where t7''&) denotes what ^''co becomes, on substituting u's for x's, and J. now 
represents 

d d d 

Wi -J h Ms T— + . . . + M„ -J— . 

aoi da^ dttn 

This expansion stops spontaneously at the (r+ l)th term, because %*■&) is 
only of r dimensions in x^, x.^... ««. 

Applying now theorem (1), we obtain 
■X^a = Vm + r (t - r + 1) V'-'^ae + \r{r- 1) {(t - r + 1) (t - r + 2)} f/''-''(»6^ ' 
+ ... + {(t-r + l)(f-r-+2)...tla,e'-. (2) 

In using this theorem in the course of the ensuing pages, it will be found 
convenient to assign to e a specific value, and I shall suppose it equal to 

Xfi . . 

— ; this gives 
an ^ 

= --^ 
an 



Un=Xn 


an 

— — Xn 

an 


= 0. 





And inasmuch as the TJ symbol now contains a^, a^, ...an, so that JJU'' no 
longer equals C/'''+\ I shall write Ur for U^. Theorem (2) will thus assume the 
form 

X^a>= Uroy+r{i - r + 1) Ur-^a}~ + ^r (r - 1) (t - r + 1) (t - r + 2) Ur-^(o{^^ 

+ ...+ {(,_r+l)....}a,(j?Y, (3) 



30] of Homogeneous Functions. 167 

where Ur for all values of r denotes what 

d d d 

' dai ' da^ ' ' ' ""' da^-i 

becomes, on substituting u^, Mj, ■■■Un-i for x^, x„,... Wn-i, after the processes 
of derivation have been completed : this it is essential to observe, because 

Ui, U2, ... Un-i now involve Oj, a,, ... a„_i, a„. The term «„ -j — is omitted from 

dan 

the symbol of linear derivation, because in the substitutions »„ will be 

replaced by zero. 

As an example of this last theorem, take 

(o = a' + b^ + (f + kabc ; 
then 

Xo) = ^a-x + 2h'^y + Zc-z + kbcx + kcay + kahz, 

%-&) = Qaa? + 6by"' + Qcz^ + 2kcxy + 2kayz + 2kbzx, 

j^u, = Qa? 4- Qy» + 6^3 + Qkxyz. 

Uias = 3a= \x-—\ + 36= iy- — \ + kbc (« - — j + kca [y - —] , 



C7s 



('41 



and it will be found that the equations given by theorem (.3) are satisfied, 
namely 

VO) = U(i) + 3-0), 

y=&) = fToO) + 2 . 2 - CTeo + 2 . 3 ^" ft), 

X='w = f7s(B + 3 - f/^M + 3 . 1 . 2 ^' f7&) + 1 . 2 . 3 1 Q,. 

Probably, as this theorem is of rather a novel character, the annexed 
sketch of a somewhat different course of demonstration may be not un- 
acceptable to my readers. 

We have 

( d d d \ 

and by the well-known law for homogeneous functions, 
d d 



i=fai 



dai ' "^ doz '" " dan 



168 On certain General Properties [30 

Hence 

/ Xn\ _ ( d d d 

V anJ \ doi ~ da.2 '" ""^ dan-i 

Hence 



&c. = &c. 

But in performing the process indicated by the several factors it must be 
carefully borne in mind that UUr is not = Ur+i ', this would be the case were 

it not for the terms ^ «„, ^ a;„, &c., which enter into u^, ii^... m„_i. But 

dn dn 

on account of these terms, we have 

,T„ / d d d \ f d d d 

UU,.Q)=\ih-^ — \-^h^ — h . . . + Un-i -i Wi J 1- w» :j — !-•••+ Wn-i j — 

V acti doo dan-J \ da-^ da^ dan-i. 



r Xn{ d d d 

y^+ifi) — 1 \Ui -; h M2^ 1- . . . + Mn-i ^ — 



n ^ ^ W/ U- O/jj 

ence 

TJU,.(0 = Ur+iCO — r~ U,.(o. 

(In 

Let — be called e ; we find 
X = U+ie, 

= UU + (2t -l)eU+{i- 1) ie^ 
= U, + 2(i-l)eU+(o-l)ie"-; 

= UU., + 2{i,-l)eUU + {i,-l)ie^U 

+ {l - 2)eU, + 2{i - ■2){c -1) e-'U + {t - 2)((, - 1) oe' 

= [/"a + 3 (t - 2) ef/^ + 3 (t - 2) (t - 1) e^f/" + (i - 2) (t - 1) ie>. 



30] of Homogeneous Functions. 169 

The same process being continued will lead to results identical with those 
previously obtained and expressed in theorem (3). 

The expansion of y^, treated according to this second method, appears to 
require the solution of the partial equation in differences 

ar+i, j+i = a,r, s+i + (f - 2r) a^, s, 
«o, s being given as unity for s = 1 and as zero for all other values of s. 

It is probable however that the solution of this equation might be evaded 
by some artifice peculiar to the particular case to be dealt with. I do not 
propose to dwell upon this inquiry, which would be foreign to the object 
of my present research. It may however not be out of place to make the 
passing remark, that the equations expressing -y^ in terms of powers of JJ 
admit easily of being reverted, as indeed may the more general form 

Xr = M;- + e^ti^_i + - — - er€r-iUr-2 + &C. 

which becomes the equation of formula (3), on making 

er = r{i,-r 1 —r) — , ■Xr = %'''». and ii,. = tJ^w ; 

for let U,. = 6] 6, ■ ■ • fr'^rj 



then 

whence Vr = e '^^ y, 



2/. = ^. + ^,_, + |^ + ^^ + &c.; 



and therefore u^ = %r — er%)— i + |e,-er-iXr-2 + &C- 

Thus we obtain, from equation (3), 

Ur(o = yjco — ?'(t — r + 1) %''~^(B — + &c. 

As a first application of theorem (3), I shall proceed to show how 
Joachimsthal's equation to the surface drawn from a given point (a, /8, 7, S) 
through the intersection of two surfaces </> («, y, z, <) = 0, 6 («, y, z, t) = 0, may 
be expressed under the explicit form of the equation to a cone. 

The equation in question is obtained by eliminating X between 
^X™ + %^X'«-i + ^— 2 x'^X"*-- + &c. = 0, 

^X- + x^X"^^ + ~ X^^X'"-^ + j-i^ x'S^""-' + &c. = 0, 



170 On certain General Properties [30 

where 

By theorem (3), these two equations, on writing — = e, become 

+ { U"-cf> + 2 {m -l)U(l>e + (m - 1) mtjie-} ^^ + &c. = 0, 

(9x" +{ue+ nde} x"-' + ( we + &c.) '^+ {we + 3{n-2) wee 

+ S{n-2){n-l) Ude' + {n - 2) {n - 1) ne'} j^^ + &c. 
Now on writing X = fi — e, these equations take the forms 

<^/i™ + UrPfi""-' +Wcj) ^' + &c. = 0, 

6i/i,» + cr6'/a"-i +we '^ + &c. = 0, 

as is easily seen by substituting back X + e in place of /i. Consequently 
e no longer appears in the coefficients of the terms of the equations between 
which the elimination is to be performed, and the resultant will accordingly 
come out as a function only of <f>, U<f), U'^, &c., that is, of a, /3, j, S,. 
and of 

a;-^t v-^t z-'^t 

showing that the equation in x, y, z, t, is of the form of that to a cone, as we 
know d priori it ought to be. Precisely a similar method may be applied to 
the elucidation of the corresponding theorem for a system of rays drawn from 
a given point through the locus of the intersection of two curves. 

Before entering upon some further and more interesting applications of 
theorem (3), it will be convenient to explain a nomenclature which has been 
emjjloyed by me on another occasion, and which is almost indispensable in 
inquiries of the nature we are now engaged upon. Homogeneous functions 
may be characterized by their degree, by the number of letters which enter 
into them, and lastly, by the lowest number of linear functions of the letters 
which may be introduced in place of the letters to represent such functions. 
Any such linear function I designate as an order, and am now able to dis- 
criminate between the number of letters and the number of orders which 
enter into a given function. The latter number, generally speaking, is the 
same as the former ; it can never exceed it, but may be any number of units 
ess than it. 



30] 



of Homogeneous Functions. 



171 



I need scarcely observe that a pair of points becoming coincident, a conic 
becoming a pair of lines, a conoid becoming a cone, and so forth, for the 
higher realms of space, will be expressed by the homogeneous function of the 
second order which characterizes such loci*, losing one order, that is, having 
an order less than the number of letters entering therein. Calling such 
characteristic 4){x, y, z ... t), it is well known that the condition of such loss 
of an order is the vanishing of the determinant 

daf ' dxdy '" dxdt 
d-(f> d^<i) d'<f) 
dydx' dy^ '" dydt 



d^(t) d-4> d-<f) 
dtdx' dtdy "' dt^ 

A conoid becoming a pair of planes, a cone becoming a pair of coincident 
lines, a pair of points becoming indeterminate, will, in like manner, be 
denoted by their characteristic losing two orders, and so forth, for the 
higher degrees of degradation. In like manner, in general, a homogeneous 
function of three letters of any degree losing an order, typifies that the 
locus to which it is the characteristic will break up into a system of 
right lines. 

Now let 0) be a homogeneous function of a, /3, 7 . . . S, and suppose that we 
have the equations to = 0, %&) = 0, j^a = 0, where ;j^ as above 

d 



d d 



doL^^d^'^^d<y^-'^*dh- 



I say that on eliminating any of the variables x,y,z...t between the second 
and third of the above equations, the resulting equation will be of one order 
less than the number of letters, that is, the expulsion of one letter will be 
attended by the expulsion of two orders. 
For we have, by theorem (3), 

VCt) = U(0 + 2 — (B = 0, 



i = P'oO) + 2 



and by hypothesis 
Hence we have also 



— iJa) + 2 — 



(B = 0, 



= 0. 



and since Ua, U^fo contain one order less than the number of letters in 

* If 17=0 is the equation to any loeus, U may be said to characterize the same, or to be its 
characteristic. 



172 On certain General Properties [30 

«i), the resultant of the elimination between them will contain two orders 
less than the number of letters in w ; and consequently, whichever of the 
letters x, y, z ...t we eliminate between ^&) = and j^w = 0, provided that 
<» = 0, the resultant equation will contain one order less than the number of 
letters remaining. 

Thus we see how it is that the taugent line to a conic meets it in two 
coincident points, the taugent plane to a conoid in two intersecting lines, 
and so forth, for the higher regions of space*. For instance, if we take 
<o (x, y, z, t) = 0, the equation to a conoid, and a, jS, y, B, the coordinates to 
any point therein, we shall have a) (a, /3, 7, S) = 0, 

d d d d\ ^, ^ . ^ 

'dS + 2/^ + "^ + *^)"'*^"*^''^'" = ^' 
and a> (x, y, 2, t), that is, x''^ = *^' 

X, y, z, t representing the coordinates of any point in the intersection of the 
conoid by the tangent plane. 

Consequently, by what has been shown above, on eliminating any one of 
the four letters «, y, z, t, the resultant function of three letters will contain 
only two orders, and will thus represent a pair of lines, real or imaginary, 
intersecting one another at a, j3, 7, S. 

The fact which has just been demonstrated (that the resultant of %m = 0, 
j^^cg = 0, loses an order if m = 0), indicates that on expressing one of the 
quantities x, y, z ...t in terms of the others, by means of the first equation, 
and then substituting this value in the second, the determinant of the 
equation so obtained must be zero. 

Now by virtue of a theorem which was given by me in a notef to my 
paper in the preceding number of this Joitrnal, this determinant will be equal 
to the squared reciprocal of the coefificient in the equation %&) = of the • 
letter eliminated multiplied by the determinant in respect to x, y, z ...t,\ oi 

This latter determinant is therefore zero ; but this determinant is the 
resultant of the equations 

did d \' d ( d , d \ „' 

d / d do \^ d f d d \ „ 

&c. &c. &c, &c. 

:^co = 0, thatis, (^^+y| + ...)« = 0, 

* Thus a tangential section of a hyperloous of the second degree at any point cuts it in 
two cones. 

[t p. 135 above.] 



30] 



of Homogeneous Functions. 



173 



Thus we obtain the singular law, that the symmetrical determinant 



d_d_ 
da da ' 

db da ' 

A A 
dc da ' 



A A 

da db ' 

db db "' 

dc db ' 



A^ 

da dl ' 

d^ d 
db dl "' 
d d 
' dc dl ' 



A 
da 

A 
db' 

A 
dc 



d 



d d d d d d 

dl da ' dl db ' '" dl dl ' 
d d d 

da ' db ' '" dl ' 

is zero when m is zero. 

This is easily shown independently by means of a remarkable and 
I believe novel theorem, relative to homogeneous functions. 

If to be any homogeneous function of t, dimensions of a, b, c ... I, we 
have (by Euler's theorem already repeatedly applied), remembering that 



da) da 

da ' db 



.„w .^^ do) ,, , 

-J— , -TT-'.-Tf are ail homogeneous. 





— Id) + [a 


i_^jl^ 

da db 


+ 1 


d\ 
il)'' 


= 0, 




d d . d d 
da da da db 


I ^ '^] 

da dl J 


= 0, 




d d 
db da 


, d d\ 

■ ^^Tbdl)" 


= 0, 


&c 


&c. 


&c. 






d d 

dl da " 


,d d\ 
- + ^dldl)'' 


= 0. 


Between these equations we 


may eliminate all the letters, a, b. 


c ...I, and we 


obtain the equation 










d d 
da da ' 


d d d d 
da db ' '" dadl ' 


d 
da 








d d 
db da ' 


d d d d 
dbdb"''---dbdl'"' 


d 
db'' 








d d 
dc da 


d d d d 
dc db ' '" dc dl ' 


d 
dc 


= 0. 






d d 
dl da ' 


d d d d 
dl db' ' dl dl 


d 

-dr 






d 

da 


d d 


t— 1 







174 



On certain General Properties 



[30 



As a corollary to this theorem, we see that if a) = the determinant 
obtained in the previous investigation becomes zero, agreeing with what 
has been already shown; in fact the last-named determinant is always 
equal to 



da da 



da db 



d^d 
' da dl 



d d^ 
dl da 



x:«. Ti 



dldh 



d d 
dl dl 



This remarkable theorem, which I have communicated to friends nearly 
a twelvemonth back, is here, I believe, published for the first time*. 

Suppose next that m {x, y, z) is the characteristic of a line of any degree, 
to which a tangent is drawn at the point a, /3, 7, using f7 in a manner corre- 
spondent to its previous signification to denote 



a \ d 
y J da 



+ [y-y' 



/3 \ d 



and understanding a (a, /3, 7) by w, we have for determining the point of 
intersection, a> = 0, %o) = 0, %"&> = ; and consequently, by aid of our theorem 
(3), we shall obtain 

&) = 0, 



1 



= 



By means of the two latter equations, we obtain 



-1=0, 

7/ 



G[y 



= 0, 



Thus let 2 be a homogeneous function in x and y of 1 dimensions, and let 



dz dz d?z d-z 

dx' dy' dx^' dxdy' 



(Pz_ 
dy^' 



1)6 called p, q, r, s, t; we shall find 



r, », p 




s, t, q 


= 0, 


P, 1, ^-33" 




I - 1 rg2 - 2p?s + tp^ 



30] of Homogeneous Functions. 175 

where i''and G are each of only (n — 2) ditDensions, and serve to determine 
the intersections of the tangent with the curve, extraneous to the two 
coincident ones at the point of contact. 

Again, suppose that w is a function of any degree of any number of 
letters a, /3, 7, &c., and that we have given eo = 0, ;;^q) = 0, %^q) = 0, . . . t^'^o) = ; 
it is evident from our fundamental theorem that these equations may be 
replaced by 

&) = 0, ?7ift) = 0, U^(o = 0, ... Uma> = (i; 

and consequently that the expulsion of {m — 1) letters, by aid of the last m 
of the given equations, will be attended by the disappearance of m orders, or, 
in other words, the resultant will be minus an order, that is, will have one 
order less than the number of letters remaining in it. 

In applying to space conceptions the preceding theorem, it will be con- 
venient to use a general nomenclature for geometrical species of various 
dimensions. 

Thus we may call a line a monotheme, a surface a ditheme, the species 
beyond a tritheme, and so on, ad infinitum. 

A system of points according to the same system of nomenclature would 
be called a kenotheme. 

An w-theme has for its characteristic a homogeneous function of (n + 2) 
letters. 

Again, it will be convenient to give a general name to all themes ex- 
pressed by equations of the first degree. Right lines and planes agree in 
conveying an idea of levelness and uniformity ; they may both be said to be 
homalous. I shall therefore employ the word homaloid to signify in general 
any theme of the first degree. 

Now let a>{oc, y, z ... t) be the characteristic to an ji-theme of the nth. 
degree. 

The number of letters x,y,z...t is {n+ 2). 

As usual, let as represent o) (a, y8, 7 . . . 8), and suppose 

o) = 0, ^(u = 0, j^co = ... ')(^a) = 0, 

and consequently 

On eliminating (n - 1) letters between the n last equations, the resulting 
function will be of three letters but of only two orders, and of the 1 . 2 . 3 . . . n 
degree. Hence we see that at every point of an n-theme of the wth degree, 



176 On certain General Properties [30 

and lying in the tangent homaloid thereto, 1.2.... re right lines may be 
drawn coinciding throughout with the «-theme. 

Thus one right line can be drawn at each point of a line of the first 
order lying on the line ; two right lines at each point of a surface of the second 
order lying on the surface; six right lines at each point of a hyperlocus of the 
third degree, and so forth. 

It is obvious that a surface may be treated as the homaloidal section of 
a tritheme, just as a plane curve may be regarded as a section of a surface. 
I shall proceed to show upon this view, how we may obtain a theorem given 
by Mr Salmon for surfaces of the third degree of a particular character from 
the law just laid down, according to which a tritheme of the third degree 
admits of six right lines being drawn upon it at every point*. 

Let (o (x, y, z, t, u) be the characteristic of any tritheme of the third 
degree ; a, /8, 7, S, e, coordinates to any point in the same. Then 
o) (a, /3, 7, 8, e) = 0, and the equation to the tangent homaloid will be 
')((i) (a, /3, 7, S, e) = 0, and the equation to the polar of the second degree 
to the given tritheme in relation to the assumed point as origin, (that is, 
the infinite system of homaloids that may be drawn fi-om the point to 
touch the tritheme), will be y^w (a, /S, 7, S, e) = 0. 

But the section of any polar through its origin is the polar of the section 
to the same origin ; hence the polar to the intersection of co (x, y, z, t, u) = 0, 
with ^w (a, /S, 7, S, e) = 0, is the intersection of -x^co = with ^- w = 0. 

The projections of these intersections upon the space x, y, z, t will be 
found by eliminating u, and getting the correspondent two equations 
between x, y, z, t. Hence we see that the projection of the latter inter- 
section upon any space x, y, z, t is & cone ; or, in other words, this 
intersection itself, that is, the polar to the intersection of the tritheme 
with its tangent homaloid, is a cone ; that is to say, the surface of the 
third degree formed by cutting a tritheme of the third degree by any 
tangent homaloid has a conical point at the point of contact ; so that 
every surface of the third degree with a conical point may be considered 
as the intersection of a tritheme of the third degree with any tangent 
homaloid theretof. 

* The reduction of any eijuatiou of the sixth degree to depend upon one of the fifth may be 
shown by Mr Jerrard's method to be equivalent to drawing a straight line upon a tritheme of the 
third degree, just as the reduction of the equation of the fifth degree to a trinomial form may be 
shown to be dependent upon our being able to draw a straight line upon a ditheme of the second 
degree. Now at every point of a tritheme straight lines may be drawn, but as they keep together 
in groups of sixes they cannot be found in general at a given point without solving an equation 
of the sixth degree. 

+ So in like manner a surface of the third degree with more than one conical point may be 
generated by the intersection of the tritheme with a pluri-tangent plane ; and so too we may get 
other varieties by taking homaloidal sections of trithemes whose characteristics are minus one or 
more orders. 



30] of Homogeneous Functions. 177 

Hence then we see, as an instantaneous deduction from our general 
theorem, that at any conical point (when one exists) of a surface of the 
third degree six right lines may be drawn lying completely upon it. This 
theorem is thus brought into an immediate and natural connexion with the 
well-known one, that at every point in a surface of the second degree, two 
right lines can be drawn lying wholly upon the surface*. 

The last geometrical application of the theorem (3) which I shall make, 
refers to the equations employed by Mr Salmon in No. XXI. (New Series) of 
this Journal, to obtain the locus of the points on any surface at which 
tangent lines can be drawn passing through four consecutive points. I may 
remark in passing that these equations may be obtained by rather simpler 
considerations than Mr Salmon has employed so to do, and without any 
reference to Joachimsthal's theorem ; for if we take ^, 77, f, 6, as the co- 
ordinates of any point in one of the tangent lines above described, and if we 
take the first polar to the surface with this point as origin, three out of the 
four original points will be found in such polar consecutive but distinct ; and 
consequently in the second polar, referred to the same origin, two will con- 
tinue consecutive but distinct, and consequently one will remain over in the 
third polar. 

Hence writing the equation to the surface w (x, y, z, t) = 0, and using D to 

denote ^-r- +''7t-+ ^j~+^;77' ^® shall evidently have 

O) = 0, (1) 

Do) = 0, (2) 

D^w = 0, (3) 

Z)^ffl = 0, (4) 

as obtained by Mr Salmon. And the same kind of reasoning precisely 
applies to the theory of points of inflexion in curves; three consecutive 
points in a right line in this case corresponding to four such in the case 
above considered. 

If now we make ? — 7 ^ = m. 



t 



' = w, 



* If we have au indeterminate system of algebraical equations consisting of one quadratic and 
another ji' function of three variables, this may be completely resolved by considering the first as 
an equation to a surface of the second degree, finding at any point thereof the two lines which 
lie upon the surface, and determining their respective intersections with the surface represented 
by the second equation. This will require therefore the solution only of a quadratic and an n" 
equation. In like manner an indeterminate system of two equations of four variables, one of the 
third and the other of the nth degree, may be completely resolved (with the aid of the theorem in 
the text) by means of two equations, one of the sixth and the other of the nth degree. 

s. 12 



178 On certain General Properties [30 

the equations (2), (3), (4), by our theorem, may be expressed in terms of 
u, V, lu, which being eliminated we obtain an equation between x, y, z, t, 
which will express the surface whose intersection with the given surface 
ft) = serves to determine the locus of the points in question. 

Hence if we proceed in the ordinary manner to eliminate two of the four 
letters, as ^ and t^, between the equations (2), (3), (4), the resultant will be of 
the form M x <^ (^, 6), where M does not contain ^, 77, ^ or 6, and where by 
the general laws of elimination ^ (f, 6) wiW be an integral function of the 
sixth degree in respect to ^,6: and it is manifest that M x cj)(^, 6) will be 
identical with the resultant of (2), (3), (4) expressed in terms of u, v, w, 
when u and v are eliminated cy-pres of an integralizing factor, showing that 
^(f, 6) is w^ integralized, that is, is equal to (t^—zdf. Consequently as M<j) 
is of the order (n - 1) 2 . 3 + (?z - 2) 1 . 3 + (?i - 3) 1 . 2, that is, lln - 18 in 
respect to x, y, z, t, it follows that M = 0, the equation to the second surface 
spoken of above, will be of the order ll?i— 24, agreeable to Mr Salmon's 
showing. 

I shall conclude this paper by showing the application of our theorem to 
the subject propounded by Mr Jerrard and Sir William Hamilton, of systems 
of equations containing a sufScient number of variable letters for effecting 
the solution without elevation of degree. 

If we have n homogeneous equations containing a sufficient number of 
letters Oi, ao... a^ to enable us to express the solution of (?i — 1) of the 
equations under the form 

a.2 =a„ + X/Sg, 



where ai, «» ... a^, A, /So ... /3m. are supposed known, and \ is indeterminate, 
it is evident that by substituting these values in the nth equation. A, may be 
found by solving an equation of the same degree as that equation contains 
dimensions of a,, Oj ... am- 

Let us then propose this question: how many letters a^, a-j.-.a^ are 
needed to obtain a linear solution of a system of n equations 

^1 = 0, ^0=0, ...^„ = 0, 
of the several degrees tj , u... 1,1, without elevation of degree ; by a linear 
solution being understood a solution under the form 

Ol = «! + '^A, 

«2 = «2 + ^02, 



ar=ar + ^^r, 



where \ is left indeterminate. 



30 J of Homogeneous Functions. 179 

Let us suppose that «!, cL,...cir, substituted respectively for a^, ao...ar, 
satisfy the given system of equations. The determination of these values 
without elevation of degree will, from what has been said before, depend 
upon the linear solution of a system of equations differing from the given 
system by the omission of any one of them at pleasure. 



Now make 



and then write 



^ d d d 

D = Oi -=- + «„ J i- ... + a^ -J— 

dai da, da^ 



Dc^i = 0, i)-<^i = ... D'-^i = 0^ 

D4>., = 0, D-^o = ... !)''</>„ = I /m 

The values of Oj, a2...ar derived from this system, say (a)i, {a\...{a)r, 
give 

ai = ai + X(a)i, a., = a„ + 'K{a\, ... ar= ar + ^(a)r, 

a solution under the required form, where X is left indeterminate. 

The solution of this new system without elevation of degree depends on 
the linear solution of all but one of them ; this excepted one may be taken 
the one whose dimensions t,. are the highest or as high as any of the 
quantities tj, ta ... t„. 

Consequently, if we use the symbol {k^, K... kr) go denote the number of 
letters required for the linear solution (without elevation of degree) of ^i 
equations of the first degree, k^ of the second, ^3 of the third, ..., ky of 
the rth, it would at first sight appear from the preceding reduction that 
we must have 

(h, h...kr) = {K„ K,... Kr-„ k;}, 

where ^i = A,'i + ^'2 + . . . + k^-i + k^, 

Kn = L+ ...+ kr-i + kr, 



Kr-i = kr-1 + kr, 
Kr' = kr - 1. 

But now steps in our theorem (3), and shows that the system {6) may be 
superseded by another, in which the variables, instead of being a-i, a«... a„, 
will be 

(h (I'll, «■' a„,...a„_i a„; 

«.« a„ On 

consequently the number of really independent variables is only (n — 1) ; we 
must therefore have 

{h,\... kr) = 1 + {K„K^... Kr'}. 

12—2 



180 General Properties of Homogeneous Functions. [30 

Since the introduction of a new simple equation is equivalent to the 
requirement of one more disposable letter, we may write the above more 
symmetrically under the form 

{k„k„... k,) = ('K„K„... K,.-i , Kr'), 

where 'K^ = 1 + ^i + A;^ + ■ . . + A;^, 

Kr'= K-l. 

By means of this formula of reduction (k^, k^ ... kr) may be finally brought 
down to the form {L), and the value of (Z) being the number of letters 
required for the linear solution of a system of L linear equations is evidently 
Z + 2. 

Thus, to determine the number of letters required for the linear solution 
of a single quadratic, we write 

(0,1) = (2) = 4. 

For two quadratics, we write 

(0,2) = (3, 1) = (5) = 7; 

for a quadratic and a cubic, 

(0, 1, 1) = (3, 2) = (6, 1) = (8) = 10 ; 
for two cubics, 

(0, 0, 2) = (3, 2, 1) = (7, 3) = (11, 2) = (14, 1) = (16) = 18. 

These results coincide with those obtained by Sir William Hamilton in 
his Report on Mr Jerrard's Transformation of the Equation of the Fifth 
Degree in the Transactions of the British Association. I have much more 
to say on the subject of the linear solution of a system of indeterminate 
equations, and am, I believe, able to present the subject in a more general 
light than has hitherto been done ; but my observations on this matter must 
be deferred until a subsequent communication. 



31. 



REPLY TO PROFESSOR BOOLE'S OBSERVATIONS ON A 
THEOREM CONTAINED IN THE LAST NOVEMBER 
NUMBER OF THE JOURNAL. 

{^Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 171 — 174.] 

The restricted space that can be spared for discussioa in these pages, 
necessitates me to compress within the narrowest limit the remarks which 
I feel bound to make on Mr Boole's extraordinary observations* in the 
February number of this Journal, on my theorem contained in the ante- 
cedent number thereoff , which statements I cannot, in the interests of truth 
and honesty, suffer to pass unchallenged. The object of that theorem was 
to shovv how the determinant of the quadratic function resulting from the 
elimination of any set of the variables between a given quadratic function 
and a number of linear functions of the same variables, could be represented 
without performing the actual elimination by a fraction, of which the nume- 
rator would be constant whichever set of the variables might be selected for 
elimination, and the denominator the square of the determinant corresponding 
to the coefficients of the variables so eliminated. The numerator itself is a 
determinant, obtained by forming the square corresponding to the determinant 
of the given quadratic function, and bordering it horizontally and vertically 
with the lines and columns corresponding to the coefficients of all the variables 
in the given linear equations. An immediate corollary from this theorem leads 
to Mr Boole's. Conversely upon the principle that " tout est dans tout " 
Mr Boole devotes a page and a half of close print merely to indicate the 
steps of a method by which from his theorem mine is capable of being 
deduced, ending with the announcement, that the numerator in question 
is equal to the quantity 

(the symbols above employed being Mr Boole's own), and concludes with 
assuring his readers that " he has ascertained that Mr Sylvester's result is 
reducible to the above form." Mr Sylvester would be very sorry to put his 

[* Camhr. and Duhlin Math. Jour. vi. (1851), pp. 90, 284.] [t P- 135 above.] 



182 



Reply to Professor Boole's Observations 



[31 



result under any such form. Mr Boole could scarcely have reflected upon the 
effect of his words when he indulged in the remark which follows — "there 
cannot he a doubt that for the discovery of the actual relation in question, the 
above theorem is far more convenient than Mr Sylvester's." Of the value to 
be attached to this assertion the annexed comparison of results is submitted 
as a specimen. 

Let the quadratic function be 

aaf + bf + cz' + dt^ + 2exy + 2ezt + 2gx2 + 2'yyt + 2hyz + I'qxt, 

and the linear functions (taken two in number) 

Ix + my + nz + pt, 

I'x + m'y + 11' z + p't. 

My numerator will be the determinant (hereinafter cited as the extended 

determinant), 

e g ri I I' 

b h y m 111 



To find the numerator of 
symbolical operator 

d 



Mr Boole's fraction, we must form the 



i„ d n d „ ^ „ » 
I'' 1- + ni- -jT + n- -J- + p^ -j-j 
da do dc ^ dd 

ci d „ d „, d ^ d „, d ^ d 

+ 2lm -r + 2np -y- + 2ln -y- + 2mp -^ + 2lp ^ + 2mn ^- 
^ de dg ^ dy ^ dh drj , 



de 
da do 



,„d ,„ d 



dc 



dd 



I ^„ , d ^ , , d „,, , d ^ , , d r,„ , d c , I d 

-\-2lm -r + 2np ^r + 2ln ^r + "m p -=- + 2t p ^rr + 2m 71 ^j- 

\ de de dg ^ dy ^ dh drj 

and after expanding the determinant hereunder written 

V 



a 


e 


.9 


e 


b 


h 


9 


h 


c 


V 


7 


e 



d 

perform the operations above indicated upon the result so obtained. 

These are the operations and processes which, on Professor Boole's 
authority, we are to accept " as without dotibt far more convenient " than 
the one simple process of forming, and when necessary, calculating the 



31] on a Theorem of Mr Sylvester's. 183 

extended determinant above given. Here for the present I leave the 
case between Mr Boole and myself to the judgment of the readers of 
this Journal. 

In the April Number of the Philosophical Magazine* , I have shown 
that the extended determinant serves, not only to represent the full and 
complete determinant of the reduced quadratic function, but likewise all 
the minor determinants thereof; the last set of which will be evidently no 
other than the coefficients themselves. For instance, in the example above 
given, if we wish to find the coefficient of x"^ after z and t have been 
eliminated, we have only to strike out the line and column eh h y m m from 
the extended determinant ; if we wish to find the coefficient of 'f, we must 
strike out the line and column a e g rj I V ; to find the coefficient of xy, we 
must strike out the line a eg 7)11' and the column ebhym m, or vice versa. 

In each of these cases the determinant so obtained is the numerator of 
the equivalent fraction ; the denominator remaining always the same function 
of the coefficients of transformation as in the original theorem. 

Again, if there be taken only one linear equation, and by aid of it x is 
supposed to be eliminated ; and if the reduced quadratic function be called 

Ly"- + Mz" + Nf + ^Pzt + 2Qyt + ^Rzy, 
the same extended determinant as before given will serve, when stripped of 
its outer border, consisting of the line and column I' m! n p', to produce 
the various equivalent fractions : thus form the square 

L R Q 
BMP 

Q P N. 

I /" 7? I 

The numerator of the fraction equivalent to _ ^, , that is, to LM — R-, 

\ K M \ 

may be found by striking out from the form of the extended determinant the 
line and column tj y edp; that corresponding to „ p , that is, LP—RQ, will 
be found by striking out the line ghcen and the column Tjyedp, or vice 
versa ; and so forth for all the first minor determinants ; and similarly the 
second minors, that is, L, M, N, P, Q, R, may be obtained by striking out in 
each case a correspondent pair of lines and pair of columns. Thus, to find 
the numerator of L the same pair of lines and columns, namely, (ghee n), 
(rj y e d p), must be elided. To find the numerator of R, the pair of lines 
(ghee n), {tj y e d p), and the pair of columns (ebhy m), {rj y e d p), or vice 
versa, will have to be elided ; and so forth for the remaining second minors. 
I may conclude with observing, that the theorem contested by Mr Boole is 
an immediate corollary from the general Theory of Relative Determinants 
alludedf to in the " Sketch " inserted in the present number of the Journal. 
[* p. 241 belo%¥.] [t p. 188 below.] 



32. 



SKETCH OF A MEMOIR ON ELIMINATION, TRANSFORMATION, 
AND CANONICAL FORMS. 

[Cambridge and Dublin Mathematical Journal, VI. (1851), pp. 186 — 200.] 

There exists a peculiar system of analytical logic, founded upon the 
properties of zero, whereby, from dependencies of equations, transition may 
be made to the relations of functional forms, and vice versa : this I call the 
logic of characteristics. 

The resultant of a given system of homogeneous equations of as many 
variables, is the function whose nullity implies and is implied by the possi- 
bility of their coexistence, that is, is the characteristic of such possibility ; 
but inasmuch as any numerical product of any power of a characteristic is 
itself an equivalent characteristic, in order to give definiteness to the notion 
of a resultant, it must further be restricted to signify the characteristic taken 
in the lowest form of which it in general admits. 

The following very general and important proposition for the change of 
the independent variables in the process of elimination, is an immediate 
consequence of the doctrine of characteristics. 

Let there be two sets of homogeneous forms of function ; 
the 1st, ^1, (/)2 ... <^„, 

the 2nd, ■^■^, ■>^„...y^n- 

Let the results of applying these forms to any sets of n variables be 
called 

then will the resultant (in respect to those variables) of 

<^.KtO. Or,)... if,)}, 



bniif.), if.;)... Or,,)}, 



32] Elimination, Transformation, and Canonical Forms. 185 

be the product of powers (assignable by the law of homogeneity) of the 
separate resultants of the two systems, 

{(</,,), (<^.)...(<^.)), 

{(to. (t.)--(t»)}. 

By means of the doctrine of characteristics the following general problem 
may be resolved. 

Given any number of functions of as many letters, and an inferior number 
of functions of the same inferior number of letters, obtained by combining, 
inter se, in a known manner, the given functions, to determine the factor by 
which, the resultant of the reduced system being divided, the resultant of the 
original system may be obtained. 

If in the theorem for the change of the independent variables both sets 
of forms of functions be taken linear, we obtain the common rule for the 
multiplication of determinants : if we take one set linear and the other not, 
we deduce two rules, namely. That the resultant of a given set of functional 
forms of a given set of variables, enters as a factor into the resultant, 

1st, of linear functions of the given functions of the given variables ; 

2nd, of the given functions of linear functions of the given variables : 

the extraneous factor in each case being a power of what may be con- 
veniently termed the modulus of transformation, that is, the resultant of 
the imported linear forms of functions. 

From the second of these rules we obtain the law first stated 1 believe 
for functions beyond the second degree by Mr Boole, to wit, that the deter- 
minant of any homogeneous algebraical function (meaning thereby the 
resultant of its first partial differential coefficients) is unaltered by any 
linear transformations of the variables, except so far as regards the intro- 
duction of a power of the modulus of transformation. This is also 
abundantly apparent from the fact, that the nullity of such determinant 
implies an immutable, that is, a fixed and inherent, property of a certain 
corresponding geometrical locus. 

There exist (as is now well known) other functions besides the deter- 
minant, called by their discoverer (Mr Cayley) hyperdeterminants, gifted 
with a similar property of immutability. 1 have discovered a process for 
finding hyperdeterminants of functions of any degree of any number of 
letters, by means of a process of Compound Permutation. All Mr Cayley's 
forms for functions of two letters may be obtained in this manner by the aid 
of one of the two processes (to wit, that one which will hereafter be called 
the derivational process), for passing from immutable constants to immutable 
forms. Such constants and forms, derived from given forms, may be best 



186 Sketch, of a Memoir on Elimination, [32 

termed adjunctive ; a terra slightly varied from that employed by M. Hermite 
in a more restricted sense. 

The two processes alluded to may be termed respectively appositional and 
derivational. The appositional is founded upon the properties of the binary 
function x^ + yrj + z^+ ... ; in which, whether we substitute linear functions 
of as, y, z, &c., or linear functions of ^, r], f, &c., in place of x, y, z, &c., or 
^, 7?, f, &c., the result is the same. 

Consequently, if we apply the form ^ to ^, rj ... ^, and take any constant 
(in respect to ^,v---0 adjunctive to 

<j>(^, V ■■■ + (o^^+yv + ■■■ + ^^+kt) t--~\ 

calling this quantity i^{x,y...z, t), the form -v/r is evidently adjunctive to the 
form ^ : and if we expand so as to obtain 

•v/f (*', y ... z, i) = T^i («, y ...z)V- -^-^^{x, y ... z) f + &c., 

it is evident i^^, -y^^, &c. will be each separately adjunctive to <^. These 
forms, when -v/r is obtained by finding the determinant in respect to ^, rj ... ^ 
of 8, are, in fact, identical with Hermite's " formes adjointes." 

The derivational mode of generating forms from constants depends upon 
the property of the operative symbol 

cJ d d 

^ = ^dx + '>d^j-^--^^dz' 

applied to a function of x, y ... z; namely, that if in 0, in place of these 
letters, we write linear functions thereof, to wit x', y' ... z', we may write 

^ = ^ d^'+'>dy^ + - + ^ d^" 

where ^', t]' ... f will be the same functions of f, ■»?... f that x', y' ...z are of 
X, y ... z. 

Suppose now, in the first place, that in regard to ^, ?; ... f, ^{x,y...z) is 
adjunctive to ;^''(^ {x,y ...z); then is the form ^ adjunctive to the form ^, for 
on changing x, y ... z to x, y ... z, 

[ <- d d . ^ dy , . , , ,, 

[^^x + ''dy^-^^dz)'^^'''y-'^ 

becomes (f' -^ +^' ^, + ...+ f A) ^(,', y ....') ; 

and consequently ■f{x,y...z) becomes -^ {x', y' ... z), multiplied by a power 
of the modulus of transformation, the modulus of that transformation, be it 
well observed, whereby od , y ■■■z would be replaced by x,y...z, and not as 
in the appositional mode of that converse transformation according to which 



32] Transformation, and Canonical Forms. 187 

x,y ... z would be replaced by *■', y' ... z'. It is on account of this converse- 
ness of the modes of transformation that the appositional and derivational 
modes of generating forms cannot except for a certain class of restricted 
linear transformations be combined in a single process. More generally, if 
instead of a single function ■)(^j>(x, y ... z), we take as manj' such with 
different indices to ^ ^^ there are variables, anil form either the resultant 
in respect to ^, v ■■■ ^> °^ '^^y other immutable constant in regard to those 
variables, (presuming in extension of the hyperdeterminant theory and as no 
doubt is the case, that such exist), every such resultant or other constant will 
give a form of function of x, y ... z adjunctive to the given form (f). 

It may be shown that every such resultant so formed will contain ^ as a 
factor. 

Again, in the former more available determinant mode of generation, if 
we take the determinant in respect to ^, ■»;... ^, it may be shown that all the 
adjunctive functions so obtained will be algebraical derivees of the partial 
differential coefficients of (/> in respect to «, y ... z: that is to say, if these be 
respectively zero, all such adjunctive functions so derived, as last aforesaid, 
will be zero, or in other words, each such adjunctive is a syzygetic function of 
the partial differential coefRcients of the primitive function. 

To Mr Boole is due the high praise of discovering and announcing, under 
a somewhat different and more qualified form and mode of statement, this 
marvel-working process of derivational generation of adjunctive forms. I was 
led back to it, in ignorance of what Mr Boole had done, by the necessity 
which I felt to exist of combining Hesse's so-called functional determinant, 
under a common point of view with the common constant determinant of a 
function ; under pressure of which sense of necessity, it was not long before 
I perceived that they formed the two ends of a chain of which Hesse's end 
exists for all homogeneous functions, but the other only when such functions 
are algebraical. 

In fact, if we give to r every value from 2 upwards, the successive 
determinants in respect to f, t; ... ^ of 

5, d d <. dV , , s 

,^^+''^+-+y •^(^'^'^^' 

will produce the chain in question, which, when (j) is algebraical and of 
n dimensions, comes to a natural termination when r = n — I. The last 
member of and the number of terms in this chain are identical with the 
last member of and the number of terms in Sturm's auxiliary functions, 
when the variables are reduced to two. There is some reason to anticipate 
that this chain of functions may be made a\ ^liable in superseding Sturm's 
chain of auxiliaries ; and if so, then the fatal hindrance to progress, ari-sing 
from the unsymmetrical nature of the latter, is overcome, and we shall be 



188 Sketch of a Memoir on Elimination, [32 

able to pass from Sturm's theorem, which relates to the theory of Keno- 
themes, or Point-systems, to certain corresponding but much higher theories 
for lines, surfaces, and n-themes generally. 

The restriction of space allowed to me in the present number of the 
Journal will permit me only to allude in the briefest terms to the theory 
of Relative Determinants, which, as it will be seen, plays an important part 
in the effectuation of the reductions of the higher algebraical functions to 
their simplest forms. Nor can the effect of the processes to be indicated be 
correctly appreciated without a knowledge of the circumstances under which 
the resultant of a given system of equations can sink in degree below the 
resultant of the general type of such system. Abstracting from the case 
when the equations separately, or in combination, subdivide into factors, this 
lowering of degree, as may be shown by the doctrine of characteristics, can 
only happen in one of two ways. Either the particular resultant obtained is 
a rational root of the general resultant, or the general resultant becomes zero 
for the case supposed, and the particular resultant is of a distinct character 
from the general resultant, being in fact the characteristic of the possibility 
not of the given system of equations being merely able to coexist (for that is 
'already supposed), but of their being able to coexist for a certain system of 
values other than a given system or given systems. Such a resultant may be 
termed a Sub-resultant; the lowest resultant in the former case may be 
termed a Reduced-resultant. The theory of Sub-resultants is one alto- 
gether remaining to be constructed, and is well worthy equally of the 
attention of geometers and of analysts. 

As to the theory of Relative Determinants, the object of this theory 
is to obtain the determinant resulting from eliminating as many variables 
as can be eliminated, chosen at pleasure from a set of variables greater in 
number than the equations containing them ; and the mode of effecting 
this object is through the method of the indeterminate multiplier. To avoid 
the discussion of the theory of sub-resultants and other particularities, I shall 
content myself with giving the rule applicable to the case (the only one of 
which as yet a practical application has offered itself to me in the course of 
my present inquiries) when all but one of the functions are linear. 

If U, A. L2 ... Lm be the first an w" and the others linear functions of n 
variables, and it be desired to find the determinant of the resultant arising 
from the elimination of any m out of the n variables, the following is the 
rule : 

Find the determinant, that is, the resultant of the partial differential 
coefficients in respect to the given variables, and of Xj, X.,...\,n of 

f-t-ZiXi-f Z/aX^-f- ... + Lm^^m- 



32] Transformation, and Canonical Forms. 180 

This resultant, in its lowest form, will be always a rational {n — l)th root of 
the resultant of the homogeneous system of equations to which the system 
above given can be referred as its type ; and this reduced resultant divided 
by a power (determinable by the law of homogeneity) of the resultant of 
ii, is--- Lm, when all but the selected variables are made zero, will be the 
resultant determinant required *. As regards what has been said concerning 
the reducibility of the general typical resultant in the case before us, this is 
a consequence of, and may be brought into connexion with, the following 
theorem, which is easily demonstrable by the theory of characteristics. If 
Qi. Qa ••• Qm be m homogeneous functions of m, variables of the same 
degree, r of which enter in each equation only as simple powers uncom- 
bitied with any of the other variables, then the degree of the reduced 
resultant is equal to the number of the equations multiplied by th& 
(m — 7 — l)th power of the number of units in the degree of each, 
subject to the obvious exception that when r is m, (there being in fact 
but one step from ?" = m — 2 to r — m), instead of r, (i — 1) must be em- 
ployed in the above formula. As an example of a sub-resultant as 
distinguished from a reduced-resultant, I instance the case of three 
quadratics U, V, W, functions of x, y, z, in each of which no squared 
power of z is supposed to enter : it may easily be shown by my dialytic 
method that instead of six equations, between which to eliminate a?, y-, z^, 
■ xy, xz, yz, we shall have only 5, the three original ones and two instead of 
three auxiliaries between which to eliminate «-, y^, xy, xz, yz, the apparent 
resultant is accordingly of the 9th instead of the 12th degree. But this is 
not the true characteristic of the possibility of the coexistence of the given 
systems, which in fact is zero, as is evidenced by the fact that they always do 
coexist, since they are always satisfiable by only two relations between the- 
variables, to wit a; = 0, y = Q. The apparent resultant is then .something 
different, and what has been termed by the above a Sub-resultant. 

I take this opportunity of entering my simple protest against the appro- 
priation of my method of finding the resultant of any set of three equations- 
of degrees equal or differing only by a unit, one from those of the other two^ 
by Dr Hesse, so far as regards quadratic functions, withr acknowledgment, 
four years after the publication of my memoir in the ^ jsophical Magazine : 
the fundamental idea of Dr Hesse's partial metho . identical with that of 
my general one. Still more unjustifiable is the sul sequent use of the dialytic 
principle, by the same author, equally without acknowledgment, and in cases 
where there is no peculiarity of form of procedure to give even a plausible 
ground for evading such acknowledgment. It is capable of moral proof that 

* The same method applies not only to the Final or Constant Determinant, but likewise to 
all the Functional Determinants in the chain above described, extending upwards from this to- 
the Hessian, or as it ought to be termed, the first Eoolian Determinant. 



190 Sketch of a Memoir on Elimmatio7i, [32 

what I had written on the matter was sufficiently known in Berlin and at 
Konigsberg, at each epoch of Dr Hesse's use of the method. 

I now proceed to the consideration of the more peculiar branch of my 
inquiry, which is as to the mode of reducing Algebraical Functions to their 
simplest and most symmetrical, or as my admirable friend M. Hermite well 
proposes to call them, their Canonical forms. Every quadratic function of 
any number of variables may always be linearly transformed into any other 
quadratic functions of the same, and that too in an infinite variety of ways ; 
but in every other instance there will be only a limited number of ways, 
whereby, when possible, one form will admit of being transmuted into any 
other : and with the sole exception of a cubic function of two letters, such 
transmutation will never be possible, unless a certain condition, or certain 
conditions, be satisfied between the constants of the forms proposed for 
transmutation. The number of such conditions is the number of para- 
meters entering into the canonical form, and is. of course equal to the 
number of terms in the general form of the function diminished by the 
square of the number of letters. Thus there is one parameter in the 
canonical foi'm for the biquadratic function of two and the cubic function 
of three letters, and no parameter in the cubic function of two letters. 
Hitherto no canonical forms have been studied beyond the cases above 
cited, but I have succeeded, as will presently be shown, in obtaining 
methods for reducing to their canonical forms functions with two and four 
parameters respectively. Owing to what has been remarked above, the 
theory of quadratic functions is a theory apart. Simultaneous transforma- 
tion gives definiteness to that theory, but has no existence for any useful 
purpose for functions of the higher degrees. Where the theory of simul- 
taneous transformation ends, that of canonical forms properly begins ; and 
in what follows, the case of quadratic forms is to be understood as entirely 
excluded. Such exclusion being understood, there is no difficulty in assigning 
the canonical, that is, the simplest and most symmetrical general, form to 
which every function of two letters admits of being reduced by linear trans- 
formations. If the degree be odd, say 2m + 1, the canonical form will be 

if the degree be even, say 2m, the canonical form will be 

Ml"" + u„^ -t- . . . -h it,^™ -I- K (iti ?t, . . . u^y, 
all the u's being linear functions of the two given variables. It is easy to 
extend an analogous mode of representation to functions of any number of 
letters. From the above we see that for cubic, biquadratic, and quintic 
functions of two letters, the canonical forms will be respectively 

11^ + v^, u'^ + v^ -\- Ku-v', ti? ■^-tF + vf, 
with a linear relation in the last-named case between u, v, w. 



32] Transformation, and Canonical Forms. 191 

First as to the reduction of any 4" function to Cayley's form 
uf' + v^ + K'U-v-. 

This may be effected in a great variety of ways, of which the following is not 
the simplest as regards the calculations required, but the most obvious. Let 
the modulus of transformation, whereby the given biquadratic function, say 
F{x,y), becomes transmuted into its canonical form, be called M; let the 
determinant of F be called D^, and the determinant of the determinant in 
respect to ^ and i) of 

which latter, for brevity's sake, may be termed the Hessian of F, (although 
in stricter justice the Boolian would be the more proper designation), be 
called D„. Then, by examining the canonical form itself (which is as it 
were the very palpitating heai^t of the function laid bare to inspection), 
we shall obtain without difficulty the two equations 

(1 - 9m0^ = If ^^A^e- 

m^ (1 - 9m^)= (m= - 1)^ = M^D^ ^^^ . 

Eliminating the unknown quantity M, we obtain 

m^(m^— 1)^ m^ — m , 

{l-dnv'f ' l-9m= ' 

where c is a known quantity. 

This cubic equation for finding m is of a pecu.iar form ; it being easy to 
show a priori, by going back to the canonical form, that its three roots are 
m, 6 (m), 6^ (m), where 

(m) = „ , 

6 being a periodical form of function such that 6^ (m) = m. 

This it is which accounts for the simple expression for m, that may be 
obtained by solving the cubic above given. A better practical mode is to 
take, instead of the determinant of the given function and its Hessian, the 
two hyperdeterminants and eliminate as before : a cubic equation having 
precisely the same properties, and in fact virtually identical with the former, 
will result. When m and consequently ilf are found, there is no difficulty 
whatever, calling the given function F and its Hessian H {F), in forming 
linear functions of the two, as 

(f>{m)F+ f{m)H{F)] 
4>,{m)F + ^{r,0n}ff{F)]' 
which shall be equal to, that is, identical with, (w' + v^y and u^v-, whence 
u and v' are completely determined. 



192 Sketch of a Memoir on Elimination, [32 

Another and interesting mode of solution is to take, besides the given 
function F and its Hessian, either the second Hessian or the post-Hessian 
of the given function, by the post-Hessian understanding the determinant in 
respect of ^ and 77 of 

(f.d dy ^ 

[^d. + ^dy)^-- 

any three of the four functions will be linearly related, and it may be shown 
that, calling either the second Hessian (that is, the Hessian of the Hessian) 
or the post-Hessian H', we shall have 

H'(F) + aH(F) + bF=0, 

where a and b will be rational and integer functions of the coefficients of 
F, and numerical multiples of two quantities R and S, such that the 
determinant of F will be equal R^ + S"^; and this, be it observed, without 
any previous knowledge of the existence of these hyperdeterminants R 
and S. 

If now we go to Hesse's form for a cubic function of three letters, we 
shall find that precisely similar modes of investigation apply step for step. 
Calling the function F and its Hessian H{F), and the post-Hessian or second 
Hessian at choice H' (F), we shall find 

H' (F) + mSH(F) + nR^F = 0, 

where m and ?i are numerical quantities and R^ + S- equal the determinant 
of F. It is interesting to contrast this equation with the one previously 
mentioned as applicable to the 4" functions of two letters, namely, 

H' (F) + mRH (F) + nSF = 0. 

In both instances there is no difficulty in assigning the relations between 
the original R and S, and the R and S of any adjunctive form. All 
Aronhold's results may be thus obtained and further extended without 
the slightest difficulty. As regards the equation for finding the parameter 
in Hesse's canonical form for the cubic of three letters, this will be of the 
4th degi-ee in respect to the cube of the parameter, and the roots will be 
functionally representable as 

X ; 6 (x) ; (p (x) ; ■\}r (x), 

where 6- {x) = <f>^ (x) = '^'^{x) = x; 

0<f>{x)= (1)6 (x) = yjr (x), 

(pyfr (x) = ^fr(f> (x) = 6 {x), 

yjre (x) = dyfr {x) = ^ («) ; 

owing to which property the equation is soluble under the peculiar form 
observed by Aronhold. 



32] Transformation, and Canonical Forms. 193 

I pass on now to a brief account of the method, or rather of a method 
(for I doubt not of being able to discover others more practical), of reducing 
a function of the 5th degree of two letters (say of x and y) to its canonical 
form u^ + v^ + iu\ subject to the linear relation au + bv + cw = 0, where the 
ratios a :b : c, and the linear relations between m, v, w and the two given 
variables are the objects of research. Here I have found great aid from the 
method of Relative Determinants; and I may notice that the successful 
application of more compendious methods to the question would be greatly 
facilitated were there in existence a theory of Relative Hyperdeterminants, 
which is still all to form, but which I little doubt, with the blessing of God, 
to be able to accomplish. It may some little facilitate the comprehension of 
what follows, if c be considered as representing unity. 

Calling as before the given quintic function F, the modulus of transforma- 
tion M, the Hessian and post-Hessian of F, H and H', and its ordinary or 
constant determinant B, we shall find 

a^w'w' -f- hHifiu^ + c'li'v^ = M-H, 

and P,P,P,P, = M'H', 

where Pi = a^vw + b^wu + c^uv, 

P2 = a^vw — b^wu — c^uv, 

P3 = — a"^vw + b'^wu — c^uv, 

Pi= — a^vvj — b^wu + c^ uv ; 

also D = M^" multiplied by the product of the sixteen values of 

a*-F&*(l)i-|-c*(l)i. 

From the above equations it may be shown that R' (a known function of 
the 8th degree of the given variables x, y) must be capable of being thrown 
under the form 

L {{x - a^y) (x - Uojy) x {x - a^y) (x - a^y) 

x(x- chy) (x - a,y) x(x- a.y) (x - a^y)}, 

where («! — a^Y x (ug — a^)- x {a^ — a^f x (Oy — a^f 

so that ^ is a known quantity*. Accordingly the said equation of the 8th 
degree, considered as an algebraical equation in - , may by known methods be 

* Or in other words, the post-Hessian determinant of a given function in two letters of the 
second degree, may be divided into four quadratic factors in such a way that the product of the 
determinants of these several factors shall be equal to the determinant of the given function. 

s. 13 



194 Sketch of a Memoir on Elimination, [32 

found by means of equations not exceeding the 4th or even the 3rd degree : 
in fact, to do this it is only necessary to form the equation to the squares of 

the differences of the roots of - in the equation H' -^y^ = 0, which new equa- 
tion will be of the 28th degree. If we then form two other equations of the 
378th degree, one having its roots equal to «/-^ multiplied by the binary 
products of the twenty-eight roots of the equation last named, the other to 
V-fiT multiplied by the reciprocal of such binary products, the left-hand 
members of these two equations expressed under the usual form will have 
a factor in common, which may be found by the process of common measure 
and will be of the 6th degree, whose roots consisting of three pairs of 
reciprocals may be found by the solution of cubics only. 

In this way, by means of cubics and quadratics, 

(cii-aa)-, (as -0.4)-, (aj-ae)-, {ar,-a^y, 
can be found, which being known, 

a^a„, a-iUi, a^a^, Oya^, 

can be determined in pairs by means of quadratics from the equation 
H' -=ry'^ = 0. This being supposed to be done, we have 

P3 = hL„ 

P, = kL„ 
where i,, Zo, is, L^, are known quadratic functions of x and y. To 
determine the ratios of/, g, h, k, we have three equations* obtained from 
the identity 

fL, + gL, + hL, + kL,{= P, + P, + P,+ P,) = 0; 

f:g:h:k being known, fL^ : gL,, : hL^ : kL^ are known ratios. 
But P^ + P2=2a^vw, 

P, + Pi= 2b^wu, 

P, + Pi = 2c^UV. 

Hence a^viu = \P, 

b^wu = XQ, 

c^uv = \R, 
where P, Q, R are known quadratic functions of x, y. 

* For we must have the coefficients of x^, xy and y'^ in 

of all them zero. 



32] Transformation, and Canonical Forms. 195 

Hence a:b:c may be found by means of the identical equation 
aHvV + h-ii?i(f + c-vhi? = H (F), 

whereby the ratios a~ - :b'^ : c~^ can be obtained without any further 
extraction of roots, showing that there is but one single true system of 
ratios a^:b'^:c'^ applicable to the problem; a:b:c being thus found, X is 
easily determined, and thus finally u, v, w are found in terms of x and y*. 

I have little doubt that a more expeditious mode of solution than the 
foregoing-f will be afforded by an examination of the properties and relations 
of the quadratic and cubic forms, adjunctive to the general quintic functions, 
and indeed to every (4n+ 1)" function of two letters hereinbefore adverted to. 

SuflScient space does not remain for detailing the steps whereby the 
general cubic function oi four letters may, by aid of equations not trans- 
cending the fifth degree, be reduced to its canonical form iv' + v^ + iv^ + p^ + q\ 
wherein u, v, w, p, q are connected by a linear equation 

au + bv + cw + dp + eq = 0; 

the four ratios of whose coefficients a:b:c:d:e give the necessary number 

4 5.6 

^-^ — 4^ parameters furnished by the general rule. Suffice it For the 

present to say, that the analytical mode of solution depends upon a cir- 
cumstance capable of the following geometrical statement : Every surface 
of the 4th degree represented by a function which is the Hessian to any 
given cubic function whatever of four letters, has lying upon it ten' straight 
lines meeting three and three in ten points, and these ten points are the only 
points which enjoy the following property in respect to the surface of the 3rd 
degree denoted by equating to zero the cubical function in question, to wit, 
that the cone drawn from any one of them as vertex to envelop the surface, 
will meet it not in a continuous double curve of the 6th degree, but in two 
curves each of the 3rd degree, lying in planes which intersect in the ten lines 
respectively above named ; so that to each of the ten points corresponds one 
of the ten lines : these ten points and lines are the intersections taken 
respectively three with three, and two with two, of a single and unique 
system of five principal planes appurtenant to every surface of the 3rd 
degree, and these planes are no other than those denoted by 

■u = 0, v = 0, w = 0, p = 0, q = 0. 

* The problem thus solved may be stated as consisting in reducing the general function 
ax^+bx^ +cxh/^+dxh/^ + exy'^+fy^ to the form 

{Ix+my)^ + {I'x + m'yY + (l"x + m"yY. 

t The coefficients in the reducing recurrent equation of the 6th degree in the process above 
detailed may rise to be of 541632 dimensions in respect to the original coefficients in F. 

13—2 



196 Sketch of a Memoir on Elimination, [32 

I have found also by the theory of Sub-resultants, that the analogy 
between lines and surfaces of the third degree, in regard to the existence 
of double and conical points, is preserved in this wise : that in the same 
way as a double point on a curve of the 3rd degree commands the existence 
of a double point on its Hessian, so does a conical point in a surface of the 
3rd degree command over and above the 10 necessary, and so to speak 
natural conical points, at least one extra, that is to say an 11th conical 
point on its Hessian. And here for the present I must quit my brief and 
imperfect notice of this subject, composed amidst the interruptions and 
distractions of an oificial and professional life. 

Observation. It may be somewhat interesting and instructive to my 
readers, to have a table of the successive scalar* determinants of a quintic 
function of two letters presented to them at a single glance. Preserving the 
notation above [page 193], we have the following expressions: 

The given function = u^ -\-v^ + vf, 

its Hessian = M^(aVt/f + Jfvfv? + c^mV), 

its post-Hessian = M^x the product of the four forms of 

a^-vw + h^ {ly^wu + c^ {ly^uv; 

its prseter-post-Hessian = M^'^ x the product of the nine forms of 

and the final determinant = if '» x the product of the sixteen forms of 

The success of the method applied depends (as above shown) upon the 
fact of a certain function of the roots of the post-Hessian (which is an oetavic 
function of ihe variables) being known, which fact hinges upon the circum- 
stance that 

{MJ X {M"-Y = iU^». 

P.S. I have much pleasure in subjoining the cubical hyperdeterminant 
of the 12th degree function of two letters, worked out upon the principle of 
Compound Permutation hinted at in the foregoing pages, for which 1 am 
indebted to the kindness and skill of my friend Mr Spottiswoode. 

* By which I mean the determinants in respect to f, 17 of 



32] Transformation, and Canonical Forms. 197 

The function being called 

12 11 

aa;'^ + 126a;"2/ "I s — caj'^y' + &c. . . . + ly^~, 

the following is* its cubical hyperdeterminant : 

agm — Qahl + 15aik + lOaj^ — 66/m, 

- Mbhk + SObgl + 2.0bij - 24c/^ + 1 Ucgk, 

- 145ci= + 50chj + IScem + 20cgi + 20ch\ 

- WOdgj + 28Qdhi + 20del + oOdfe + lOd-k, 
+ d85egi - r35e^k - 290eh- + 105/gh, 
-S^O/H-BOf. 

Mr Spottiswoode will I hope publish the work itself in the next number 
of the Journal, in which I shall also show how the hyperdeterminants of the 
cubical function of three letters, Aron hold's (S and T, may be similarly 
obtained. 

[* See below, p. 202.] 



33. 

ON THE GENERAL THEORY OF ASSOCIATED 
ALGEBRAICAL FORMS. 



[Cambridge and Dublin Mathematical Journal, vi. (1851), pp. 289 — 293.] 

The following brief exposition of the general theory of Associated Forms, 
as far as it has been as yet developed by the labours or genius of mathema- 
ticians, is intended as elucidatory and, to a certain extent, emendative of 
some of the statements in my paper* on Linear Transformations, in the 
preceding number of the Journal. 

In the first place, let a linear equivalent of any given homogeneous 
function be understood to mean what the function becomes when linear 
functions of the variables are substituted in place of the variables them- 
selves, subject to the condition of the modulus of transformation (that is, 
the value of the determinant formed by the coefficients of transformation) 
being unity. 

Secondly, let two square arrays of terms (the determinants corresponding 
to each of which are unity) be said to be complementary when each term in 
the one square is equal to the value of what the determinant represented by 
the other square becomes when the corresponding term itself is taken unity, 
but all the other terms in the same line and column with it are taken zero. 
This relation between the two squares is well known to be reciprocal. Thus, 
for instance. 



a 


b 


c 




a 


/3 


7 


a' 


b' 


c' 


and 


a 


13' 


7' 


a" 


b" 


c" 




a" 


/3" 


7" 



will be said to be reciprocally complementary to one another when the two 
determinants which they represent are each unity, and when we have 
[* p. 184, above.] 



33] General Theory of Associated Algebraical Forms. 199 



h = 



1 











/3' 


7 





/3" 


7' 





1 





a! 





7' 


a" 





7" 


a 





7 





1 





a" 





7' 



1 











&' 


c' 





6" 


c" 





1 





a 





c' 


a" 





c" 


a 





c 





1 





a" 





c" 


&c. 





&c. 

Accordingly, two transformations, say of F(x,y,z) and G{u,v,w) respect- 
ively, may be said to be concurrent when in F for x, y, z, we write 

ax +hy +CZ, 
a'x +b'y +c'z, 
a"x + b"y + c"z ; 
and in G for u, v, w, we write 

au +bv + cw, 
a'u +b'v +c'w, 
a"u + b"v + c"w ; 

but complementary when for u, v, w, we write 

oai + I3v +7W, 

a'u + j3'v + y'w, 

a"u + ^"v + 7"w ; 
a, b, c, &c., a, /3, y, &c. being related in the manner antecedently explained. 

Two forms, each of the same number of variables, may be said to be 
associate forms when the coefficients of the one are functions of those of the 
other; and when it happens that the coefficients of the first are all explicit 
functions of those of the second, the latter may be termed the originant and 
the former the deriyant. 

If now all the linear equivalents of one or of two associated forms are 
similarly related to corresponding linear equivalents of the other, so that 
each may be derived from each by the same law, the forms so associated will 
be said to be concomitant each to the other. This concomitance may be of 
two kinds, and very probably, in the nature of things, only of the two kinds 
about to be described. 



200 On the General Theory of [33 

The first species of concomitance is defined by the corresponding 
equivalents of the two associated forms being deduced by precisely similar, 
or, as we have expressed it, concurrent transformations or substitutions, each 
from its given primitive. The second species of concomitance is defined by 
the corresponding equivalents being deduced not by similar but by contrary, 
that is, reciprocal or complementary substitutions. Concomitants of the first 
kind may be called covariants ; concomitants of the second kind may be 
called contra variants. When of the two associated forms one is a constant, 
the distinction between co- and contra-variants disappears, and the constant 
may be termed an invariant of the form with which it is associated*. It 
follows readily from these definitions that a covariant of a covariant and a 
contravariant of a contravariant are each of them covariants ; but a covariant 
of a contravariant and a contravariant of a covariant are each of them 
contravariants ; and also that an invariant, whether of a covariant or of 
a contravariant, is an invariant of the original function -(-. 

It will also readily be seen that as regards functions of two letters 
a contravariant becomes a covariant by the simple interchange of x, y 
with — y, X, respectively. Covariants are Mr Cayley's hyperdeterminants ; 
contravariants include, but are not coincident with, M. Hermite's formes- 
adjointes, if we understand by the last-named term such forms as may be 
derived by the process described by M. Herraite in the third of his letters to 
M. Jacobi, " Sur differents objets de la Theorie des Nombres," (which process 
is an extension of that employed for determining the polar reciprocal of an 
algebraical locus:];). M. Hermite appears, however, elsewhere to have used 

* Accordingly an invariant to a given form may be defined to be such a function of the 
coefficients of the form, as remains absolutely unaltered when instead of the given form any 
linear equivalent thereto is substituted. Of course if the determinant of the coefficients of the 
transformations correspondent to the respective equivalents be not taken unity as supposed in 
this definition, the effect will be merely to introduce as a multiplier some power of the deter- 
minant formed by the coefficients of transformation. 

t It may Ukewise be shown that linear equivalents of covariants and contravariants are 
themselves related to one another as covariants and contravariants respectively, the transforma- 
tions by which the equivalents are obtained being taken concurrent in the one case and contrary 
or reciprocal in the other ; and of course any algebraic function of any number of covariants is 
a covariant and of contravariants a contravariant. 

X This has been further generalized by me in the theorem § given in the last number of this 
Journal, where I have shown in effect that any invariant in respect to 4, 77 ... S of 

/(f, T, ... e) + {x^ + yv + ...+t8 + p)p''-\ 
(/ being supposed to be of the degree ti) is a contravariant ot f {x, y ..,t). When this invariant 
is the determinant of /, it may be shown that we obtain M. Hermite's theorem. It is somewhat 
remarkable that contravariants should have been in use among mathematicians as well iu 
geometry as the theory of numbers (although their character as such was not recognized) before 
covariants had ever made their appearance. Invariants of course first came up with the theory 
of the equation to the squares of the differences of the roots of equations, the last term in such 
equation being an invariant. I believe that I am correct iu saying that covariants first made 
their appearance in one of Mr Boole's papers, in this Journal ; but Hesse's brilliant application 

[§ p. 186 above.] 



33] Associated Algebraical Forms. 201 

the term forme-adjoiate in a sense as wide as that in which I employ 
contra variants. For instance, he has given a most remarkable theorem, 
which admits of being stated as follows : 

If we have a function of any number of letters, say of x, y, z, as 

a,«"' + 'mbx'"^~'^y + mcx"^''^z -\ ^^— ~ dx'^^'^y'- + &c., 

and if / be any invariant of this function, then will 

r ^ + " ^ db + " ^53 + "^ ^ dd H ^ 

be a " forme-adjointe " of the given function. It is perfectly true and admits 
of being very easily proved, as I shall show in your next number, that this is 
a contravariant of the given function*; but it is not (as far as I can see) a 
forme-adjointe in the sense in which the use of that word is restricted in the 
letter alluded to. If, however, we adopt as the definition of formes-adjointes 
generally, that property in regard to their transformees which M. Hermite 
has demonstrated of the particular class treated of by him in the letter 
alluded to, then his formes-adjointes become coincident with my contra- 
variants. It will thus be seen that covariants and contravariants form two 
distinct and coextensive species of associated forms, which divide between 
them the wide and fertile empire of linear transformations so far as its 
provinces have been as yet laid open by the researches of analysts. In 
your next number I propose to enter much more largely into the subject 
generally. More particularly I shall describe the new method of Permutants, 
including the theory of Intermutants and Commutants (which latter are 
a species of the former, but embrace Determinants as a particular case), and 
their application to the theory of Invariants. I shall also exhibit the con- 
nexion between the theory of Invariants and that of Symmetrical Functions, 
and some remarkable theorems on Relative Invariants -f-. 

Some of your readers may like to be informed that a Supplement to my 
last paper, under the title of " An Essay on Canonical Forms," has been since 
published J; and that I have there given a much simpler method of solution 
of the problem of the reduction of quintic functions to their canonical form 
than in the original memoir, and extended the method successfully to the 

of one from among the infinite variety of these forms to the discovery of the points of inflexion 
in a curve of the third order, in other words, to the Canonical Reduction of the Cubic Function 
of Three Letters, appears to have been the first occasion of their being turned to practical 
account. 

* This is also true if / be taken any covariant instead of an invariant of the function. 

t It will be readily apprehended that the definitions and conceptions above stated, respecting 
covariants and contravariants of two single functions, may be extended so as to comprehend 
systems of functions covariantive or eontravariantive to one another. 

X By Mr George Bell, University Bookseller, Fleet Street, [p. 203 below.] 



202 General Theory of Associated Algebraical Forms. [33 

reduction of all odd-degreed functions to their canonical form. I may take 
this occasion to state that the Lemma given in Note (B) of the Supplement, 
upon which this method of reduction is based, is an immediate deduction 
from the well-known theorem for the multiplication of Determinants. 

There is a numerical error in " The Cubical Hyperdeterminant of the 
Twelfth Degree," worked out after the method <jf commutants by Mr Spottis- 
woode, given at the end of my paper in the May Number. The correct result 
will be stated in the next number of the Journal, where I hope also to be 
able to fix the number of distinct solutions of the problem of reducing a 
Sextic Function to its canonical form 

u^ -I- d" -f i(f -\- mw-y-w^. 

For odd-degreed functions there is never more than one solution possible, as 
shown in the Supplement referred to. 

P.S. Since the above was sent to press, I have discovered an uniform 
mode of solution for the canonical reduction of functions, whether of odd or 
even degrees. The canonical form however, except for the fourth and eighth 
degrees, requires to be varied from that assumed in my previous paper. Thus, 
for the sixth degree the canonical form will be 

aifi + hif + cvf -h muviu (v— w){w — it){u — v), 

where u, v, w are supposed to be connected by the identical equation 
u + v + w = 0. And there will be only two solutions — a remarkable and 
most unexpected discovery. For functions of the eighth degree there are 
five distinct solutions, and in general there is the strongest reason for be- 
lieving (indeed it may be positively affirmed) that when the canonical form 
has been rightly assumed for a function of the even degree n, the number of 
solutions will be -^ (n -I- 2) when \n is even, but I {n + 2) when ^n is odd. It 
turns out therefore that the theory for functions of the sixth degree is in 
some respects simpler than for those of the fourth. The investigation into 
canonical forms here referred to has led me to the discovery of a most unex- 
pected theorem for finding all the invariants of a certain class, belonging to 
functions of two letters of an even degree. 



34. 



AN ESSAY ON CANONICAL FORMS, SUPPLEMENT TO A 
SKETCH OF A MEMOIR* ON ELIMINATION, TRANSFOR- 
MATION AND CANONICAL FORMS. 

Since the above paper was in print I have succeeded in obtaining a 
canonical representation of the quadratic and cubic functions adjunctive to 
the general quintic (5th degreed) functions of two letters. 

Let F the quintic function of x, y, 

= u^ + w^ + w^, 
and 

au + bv + cw = 0, 

M being the modulus of the transformation, whereby transition is made from 
x, y to u, V. Then the quadratic adjunctive is 

— [a'^vw + o*wu + (fuv\ ; 

and the cubic adjunctive is simply 

-M^{ahcyuvw]: 

Hence we can, in accordance with what I ventured to predict in the preceding 
sketch, find u, v, w, by means of a simple and practical co-process. To 
wit, call 

F=laf + bmafy + lOnaPy' + lOpx^y^ + Bqxy" + ry^. 

[* p. 184 above. See p. 201, note J.] 

t The knowledge of the existence of these lower adjunctive forms is mainly a consequence 
of Mr Cayley's splendid discovery of hyperdeterminant constants. In fact, they are respectively 

the quadratic and cubic hyperdeterminants in respect to J and ?; of = — \^'h"^ ^ ^) ^ ' 

X and y being treated as constants. 

The fortunate proclaimer of a new outlying planet has been justly rewarded by the offer of 
a baronetcy and a national pension, which the writer of this wishes him long life and health 
to enjoy. In the meanwhile, what has been done in honour of the discoverer of a new and 
inexhaustible region of exquisite analysis ? 



204 On Canonical Forms. [34 

Form the determinant 

Ix + my, mx + ny, nx + py 

mx + ny, nx +py, P^ + ^y 

nx + py, px + qy, qx + ry 

Let this cubic function, by solving it as a cubic equation, be made 
equal to 

^ («^ +/>/) (« + gy) (*■ + %). 

then 

u = k(x +fy), V = l{x+ gy), w=m {x + hy). 

By means of the identity, F = u^ +if + vf, P, w?, n^, are known by the 
solution of linear equations, and thus u, v, w, are determined by solving 
a cubic equation instead of one of the eighth degree, as in the method first 
given, and the process of canonising a quintic function is rendered practically 
possible. 

For brevity sake let c represent unity. The constant determinant of the 
cubic adjunctive will be found to be 

SM^iabcy. 

Calling, then, the cubic adjunctive of F, C (F), we have the remarkable 
equation 

C{F) 

It may also be shown that if we call the Hessian of F, H (F), we shall have 
the following equally remarkable equation : 

aH(F) = iaFxaC(F). 

Again, calling the quadratic adjunctive of F, Q (F), we shall easily find 

' + b'-+c'-)] 

aQ{F) = . 



or, if we please, 



(a? + 61 _ cl) 
(al_6t_c?)j 



^ 2a'¥-2a'c'-2¥cH' 



When u, v, w are known, a, h, c, which are the resultants of v, w; w, u; u, v 
respectively are known. But their ratios, or, if we please to say so, the ratios 
of a^ : ¥ : c^ may be found independently and very elegantly as follows : — 

Let M^" X product of the 4 forms of a* + 1*6^ + l*c* = .4, 

i/-" X product of the 16 forms of a* 4- 1*6* + l^c* = 5, 

ilf™ X a^" . 6" .c"> = C. 



34] On Canonical Forms. 205 

A, B, C are known quantities, being respectively what we have called 
dQ(F), D{F)*,iaC(F). 

It may easily be shown that 

B-A- = l2SM^w-lf& (a' + ¥+ c'). 

Hence JiPa^, M^¥, M^& are the roots of p in the cubic equation 

A, B, C, it will be observed, are independent and, as they may be termed, 
prime or radical adjunctive constants. Hitherto much mystery and un- 
certaint}' have attached to the theory of hyperdeterminants, from its having 
been tacitly assumed that they were always either of lower dimensions than 
the ordinarj' determinant, or else algebraical functions of such, and of the 
determinant. Whereas we now see that, whilst the determinant of a function 
in two letters of the fifth degree is of eight dimensions, one of its radical or 
primitive hyperdeterminants is of four, but the other of twelve dimensions. 
This is a most valuable consequence, and would seem to indicate that the 
number of radical hyperdeterminants to a function, over and above the 
common determinant, is always equal to the number of parameters entering 
into its canonical form. The importance of this ascertainment of an un- 
suspected third radical constant, adjunctive to a quintic function of two 
letters, in making to march the theory of hj^perdeterminants, can hardly 
be over-estimated. 

From the equation last given we are enabled to assign the conditions in 
order that two functions of the fifth degree may be capable of being linearl}' 
transfoi'med either into the other. For if we call F and F' two such linearly 
equivalent quintic functions, they must be capable each of being thrown 
under the same form u^ + v° + {hi + mvY, where I and m shall be the same for 
each. Consequently we must have the roots of p in the same ratio for F 
and F', which conditions ma}^ be expressed hj means of the two equations 

B-A- '_ B'-A'' 
(B - AJ - 2'' AC _ (B' - A' J - 2^'A'G' 

* More strictly speaking (and this correction should be supplied throughout in the "Sketch"), 
B is the negative determinant of IF. After finding, by the method of characteristics, or any 
special artifices, the algebraic part of the value of a resultant or determinant, a process frequently 
of some complexity remains over in assigning its numerical multipher; this part of the operation 
being analogous to that which occurs in the Integral Calculus, of determining the constant to be 
added after the general form of an integral has been determined. In the " Sketch," a correction 
for the numerical multipUer remains also to be applied to the expressions given for the successive 
Hessian determinants. 



206 On Canonical Forms. [34 

A', B', C, of course representing the same functions of the coefficients of 
F' &S, A, B, C, respectively of i''. 

The two conditions required in their simplest form are accordingly 

A_A^ 

B^B^ 

or A':B':C::A'':B'"-:G', 

that is to say, all quintic /mictions of two letters of which the determinant 
is to the subduplicate power of the radical hyperdeterminant of the twelfth 
order and to the sesquiduplicate power of the radical hyperdeterminant of the 
fourth order in given ratios, are mutually convertible. 

So for the quartic (that is, biquadratic) function of two letters, calling R 
and S the radical adjunctive constants of the second and third orders, the 
condition of convertibility between different forms of the same is, that 
-E' : 8- shall be a given ratio. And, in general, we may infer that the 
condition of convertibility between different functions of any degree is, 
that the several radical adjunctive constants of each raised respectively 
to such powers as will make them of like dimensions, shall be to one another 
in given ratios. Of course all cubic functions of two letters, according to this 
rule, are mutually convertible without any condition, they having but one 
radical adjunctive constant ; and in fact all such functions, being represent- 
able as the sum of two cubes of new variables linearly related to those given, 
are necessarily convertible. 

I have further succeeded in obtaining the canonical form of the quadratic 
adjunctive to any odd degreed function of two letters, which presents a 
wonderful analogy to the theory of relative determinants of quadratic 
functions of any number of letters, and constitutes an important step towards 
the construction of the theory of relative hyperdeterminants. 

Let a function of two letters of the odd degree m(= 2?i— 1) be thrown 
under its canonical form. 

Ml™ + U™ -(-...+ i*,™, 

and let there exist the n—2 equations, 

Oi?*! +a2«t2+... +a,iM„ = 0, (1) 

6j Z(i + 6,M2 + ■ ■ • + bnll.n =0, (2) 



?lMl + loU« + . . . + InUn = 0. (?l - 2) 

Then, if M be the modulus of the transformation which converts u^, Uo into 



34] 



Chi Canonical Forms. 



207 



X, y, and if, on making 6^, O^-.-On disjunctively equal to 1, 2...n we 
use (^,1-1, 6n) to denote in general the determinant 



F will be 



M'' 



the quadratic adjunctive of — ; — , . 

•^ m(m — 1) ... ^ 

N.B. By means of this formula, and of the theorem for finding relative 
determinants of quadratic functions, we can obtain the general canonical 
form for one set of the biquadratic adjunctive constants (hyperdeterminants 
of the fourth order in Mr Cayley's language) of any odd degreed function 
of two letters f. 

Thus, for the fifth degree, preserving the notation of the " Sketch," we 
have the biquadratic adjunctive constant 

0, c^ 6*, a 

c\ 0, a\ h 

h\ a\ 0, c 

a, h, c, 

For the seventh degree, if we suppose the function to be equal to 

u' + v' + lu'' + &, 
and 

au->rhv-\-cw-\-dQ = 0, 

a!u + h'v + c'w + d!Q = ; 

M" , • ,- , 

the biquadratic adjunctive constant will be 

determinant 

{ac — dcf, 
(be' - b'cf, 

0, 
{dc' — d'cf, 

c, 

c', 

* The condition m = 2n - 1 is only necessary in order that S/ (m"*) may be a canonical, 
because a possible and determinate, form for any given function of the mth degree. But the 
theorem in the text, so far as it serves to obtain the quadratic adjunctive of S„' («'"), is true for 
all odd values of ))!, whether greater or less than 2n- 1. 

t See Note (A) of Appendix. 



0, 


{ab' - a'by, 


(ba' - b'af, 


0, 


{ca! — c'af, 


{cb' - c'bf, 


{da! — d'af. 


(db'-d'bf, 


a, 


b, 


a, 


v. 



{cd'-c'dY 


KiJJIiC 


J. K/ 


{ad' — a'dy, 


a, 


a' 


{bd' - b'df, 


b, 


b' 


{cd' - c'df, 


c, 


c 


0, 


d, 


d! 


d, 


0, 





d'. 


0, 






208 



On Canonical Forms. 



[34 



The determinants of the Hessian, the post-Hessian, and the praeter-post- 
Hessian of F will be found (in the case of the quintic function) to be always 
multiples of powers of the determinant of the given function, and of its cubic 
adjunctive ; and I believe that in general for a function of two letters of any 
degree the determinants of all the derived forms in the Hessian scale*, will 
be necessarily algebraical functions of any two of them. 

I hope very shortly to accomplish the reduction of functions, as high as 
the seventh degree of two letters, to their canonical form, and also to present 
a complete theory of the failing or singular cases of canonical forms. 



Since the above was in print I have discovered the following 



General Theorem 

for reducing a function of two letters of any odd degree to its canonical 
form. 

Let the degree of the function be (2ft — 1) ; then its canonical form is 

Mi-"-l + M2^-i + . . . + UrC^-\ 

with {n— 2) linear relations between Mj, u^, ... Un. 

To find Ml, u„, ... Un, proceed as follows. Let the given function of the 
(2?i — l)th degree be supposed to be 

271-2 

a-.x'^'' + (2rt - 1) a,a;-"--^y + {2n - 1) — ^ — a.^cc'^^-^y^ + ... + a^y'^-^. 

Form the determinant 

a-i^x.\-a^y, anX + a^y, a^x + aty . . . anX + a^+iy 
a^x + a^y, a^x^a^y, an+iX+an+.,y 



ttnX + an+iy, an+iX + ttn+^y, a-2n-i«> + O'my 

This determinant is a function of x and y of the ?ith degree, and by resolving 
an equation of the mth degree, may be decomposed into ?i factors, say 

{l^x + m^y) {Lx + m^y) . . . {Inos + niny) ; 

* I use the term Hessian (more properly speaking the Boolian) Scale, to denote the deter- 
minants in respect of | and i; o{ i ^ -j- + tj — + &c. \ F. 

Neither Hesse, however, nor any other writer up to the present time, had thought of con- 
structing, and still less of turning to account, the functions (the first only excepted) which figure 
in this scale. 



34] On Canonical Forms. 209 

we shall then have 



'■in=Pn{ln!« + Wl^y), 



where the I's and m's are known, and the (2?i — l)th powers of the p's may 
be found linearly, by means of the identical equation X^i™"'^ = F (x, y). Thus 
for example a function of the seventh degree of two letters may be reduced 
to its canonical form 

{Ix + myj + {I'x + m!yy + (J'x + mi'yj + {l"'x + m!"yy, 

by the resolution of a biquadratic equation. My demonstration of this 
extraordinary and unexpected consequence rests upon the following lemma*, 
itself a very beautiful and striking theorem (no doubt capable of much 
generalisation) in the theory of determinants. Form the rectangular matrix 
consisting of n rows and (?i + 1) columns 

■^2) -^Zi -^4 ...-t n-\-1} 



■•n+i) -'•n+2 ' 



T 

-'-2117 



where 



Ti = a/-%'+^ + a/-%'+^ +... + <:i&l"tV 



Then all the n+1 determinants that can be formed by rejecting any one 
column at pleasure out of this matrix are identically zero. 

In order the better to realise the proof, suppose 

n = 4<, so that 2n — 1 = 7. 

Let 

F (x, y) = ai*' + Ta^x^y + ila^xY + SSaiO^y^ + Soa^x^y^ 

+ 21aeX^y^ + la^xy^ + as^/'. 
Suppose 

i' + M'' + D'+w' = i^(«, y)=G{u, v), 

at + bu = V, 

a't + h'u = w. 

Then, if M is the modulus of transition from x, y to u, v the hyper- 

* See Note (B) of Appendix. 
S. U 



210 



On Canonical Forms. 



[34 



determinant, or, to adopt my new expression, the permutant P^ (meaning 

thereby) 

aiX+a^y, a^x + a^y, a^x + a^y, aiX+ a^y 
a^x + a^y, a^x + a^y, a^x + a^y, a^x + a^y 
a^x + a^y, a^x -\-a^y, a^x + a^y, a^x + a^y 
a^x + a^y, a^x + a^y, a^x + ajy, a,x + a^y 

(Ci Cl \^ 

QjCG ^y J 
according to the principles laid down in the preceding " Sketch," be the 
product of a power of M multiplied by the corresponding adjunctive constant 

/ d d\^ 

of ( 1^ -1- + ?? -T- j G(u,v), and is therefore a multiple of the determinant 

(l + Aj)t + A„u, A^t + A^u, A^t + AiU, A^t + A^u 

A^t + AsU, Ast + A^u, Ait + A^u, A^t + A^u 

Ast+AiU, A^t+A^u, A^t + A^u, A^t + A^u 

Aj+A^u, A^t+.AsU, A^t + Ajii, Ajt + {1 + A^yii 
where 

J.i = a' + a'^ A^ = a'^b + aH', A^ =a'b- + a''b'- ...As=b' + b''. 
In this determinant the coefficient of it* is 







A„ 


A„ 


A„ 


A,- 






A„ 


A„ 


A,, 


A, 






A„ 


A,, 


A„ 


A, 






A„ 


A, 


A, 


1+A, 




which is numerically equal to 












A„ A„ 


A, 




A„ A 


4, A, 


A, 


A„ A„ 


A, 


-A, 


As, A 


., A, 




A, 


A, 


A, 




A„ A 


6> Ai 



+ A, 



A„ As, A, 
A3, Ai, Ag 
A„ A„ A, 



(1+^8 



A._, A3, A, 
A3, A„ A, 
Ai, A^, Ag 



= 0, because the second factors of the products are all zero by the lemma. 
Hence the permutaut P4 vanishes when ^ = 0, and consequently it contains 
t as a factor, and in like manner it may be proved to contain u, v, w. 

Hence t, ti, v, w are the algebraical factors of P4, and precisely the same 
proof applies to show in the case of a function in x and y, say F.n-i, of any 



I/I 



34] On Canonical Forms. 211 

odd degree (2?i — 1) whatever, that the corresponding permutant P„ will 
contain the factors Wi, «2 ... Un linear functions oi x, y, such that 

wr»-i + M »-i + . . . + M„™-' = i^2„_i 
as was to be shown. 

Whenever P„ has equal roots, this will denote either (which is the more 
general case) that the usual canonical form fails and gives place to a singular 
form, (owing to some of the coefficients of transformation becoming infinite), 
or, whicli is the more special supposition, that the canonical form becomes 
catalectic by one or more of the linear roots* disappearing. Thus in the 
cubic function, if Pj has equal roots, and consequently its determinant 
(which is coincident with that of the function itselfj vanish, then the canoni- 
cal form in general fails ; so that, for example, aa? + hx"y cannot in general 
be exhibited as the sum of two cubes : if, however, certain further relations 
obtain between the coefficients of P, the canonical form reappears catalectically, 
the function becoming in fact representable as a single cube. So, again, for 
the quintic function (referring back to the notation above [page 205]), 
if P3 have equal roots, that is if = 0, the canonical form fails, unless at 
the same time P — J.^ = 0, in which case the function becomes the sum of 
two fifth powers; but if furthermore J. = 0, then this catalectic form again 
gives place to a singular form, which, on the satisfaction of a further condition 
between the coefficients, again in its turn gives way before a (bicatalectic, 
that is) doubly catalectic form, namely, a single fifth power. 

It is remarkable, that the form to which Mr Jerrard's method reduces the 
function of the fifth degree, expressed homogeneously as aar' + hxif- + cy^, is a 
singular form, being incapable of being exhibited as the sum of three cubes ; 
such, however, is not the case with the form aa? + hafy"^ + cy^: It may further 
be remarked, that although the singly catalectic form of the quintic function is 
expressible by two conditions only, namely, C=0, P— J.-=0, it will be indicated 
by P3 (which being a cubic function of a; and y contains four terms) completely 
disappearing, so that apparently four conditions would appear to be required 
or implied. But of course these must be capable of being shown to be 
non-independent, and to be merely tantamount to the two independent ones, 
= 0, P — J.^ = 0. The theory of the catalectic forms of functions of the 
higher degrees of two variables presents many strong points of resemblance 
and of contrast to that of the catalectic forms of quadratic functions of 
several variables. 

One important and immediate corollary from the General Theorem is, 
that the constants which enter into the linear functions appurtenant to the 
canonical form of any function of an odd degree form a single and unique 
system ; or, in other words, the canonical forms for such functions are void of 

* J(i, Mj ... «„ may be termed the linear roots of the form i^oji-i' 

U— 2 



212 



On Canonical Forms. 



[34 



multiplicity, a result contrary to what might have been anticipated, and 
to what we know is the case for the canonical forms of functions of an even 



It may further be shown that if we have the (to — 2) equations 
«!«! + ajtfa + ...+«„ Mm = 0, 
6iMi + 62 "2 + . . . + 6„w„ = 0, 



ZiMj + LU2 + ... + InUn = 0, 

and call M the modulus of transformation in respect to Ui, u^, and if we 
make 

P„ = Kv^Ui ...Un, 

then 

a„ a,... an ^"'"-« 

63, 64 ... bn ^ 



ij, Ij ... In 

is equal to the product of the ^ (n — 1) factors of the form 

I, ae ...ag^ 



be. ... be, 



lev ^h ■■• ''^n- 

6^, 62 ... 6n-2 being any (w - 2) numbers out of the n numbers 1, 2, 3 ... n. 
It may hence be shown that 



Milig . . . Un — 






O Pn ^'"-^ 



in being a number which is a function of n, and which may be shown to be 
equal to i=J (a;""^ y 
of ?"-2 = 1, that is 



equal to ^ (a;""^ y + xy^~'^) -r- product of the squared differences of the roots 



m^[ ^^Z^^'-J \ ={-nr-, 



■{n-2) 



and thus 



Wi V/2 • . • lln — < 



Pn 



D means the determinant in respect to x and y. 



34] 



On Canonical Forms. 



213 



As an example of the mode of finding Mj, 1*2 ... m„, let 

then 

3a:, 2y, 2x 

Ps= 1y, 2x, 2y =1^0? -^y^x. 
2x, 2y, 2x 

Heuce 

u =fx, v= g(x + y), w = h(x — y). 

To find/, g, h, we have u^ + v^ + w^ = F, hence 

f'+g'+h'' = S; g' + h' = 2- g'-h'=l; 

whence we have 

F = af+(x + yy + (x- yf. 

Again, we find 

D (4.«5 - iy^'x) = - # X 12, 



((^ 



and accordingly 



(-3) 



x{x + y)(x-y) = 



oP. =4, 



according to the general formula above given. 

As a second example let 

F=3x'' + 42«y + 70x'y^ + lixy^ + y'' ; 
then 

3x, 2y, 2x, 2y 

2y, 2x, 2y, 2x 
2x, 2y, 2x, 2y 
2y, 2x, 2y, 2x + y 
and accordingly we shall find 

x'' + y'' + (x- yj + {x + yj = F. 



P.= 



= 4 {(c^y — xy^) = 4!xy (x — y)(x + y), 



Moreover 

and 
Thus 



a (4a;3y ~ 4<xy^) = 4', 

44-2 _ 4,2_ 



P4 ^ yop, 

tuvw V 4''~2 ' 



agreeable to the general formula. 



214 On Canonical Forms. • [34 

As a corollary to our general proposition, it may be remarked, that if 
F^n-\ be a symmetrical function of x, y of, the (2n — l)th degree, PniF^i^i) 
will be also a symmetrical function of x and y, and may therefore be resolved 
into its factors by solving a recurring equation of the nth degree, which may, 
by well-known methods, be made to depend on the solution of an equation 
of the -^wth or J (»i — l)th degree, according as n is even or odd. 

Hence the reduction of a function of two letters of the degree 4m + 1 to 
its canonical form as the sum of powers may be made to depend on the 
solution of an equation of the mth degree ; so that, for example, a symmetrical 
function of x, y, as high as the fifteenth or seventeenth degree, may be 
reduced by means of a biquadratic equation only. 

In a short time I hope to present to the public a complete solution 
of the canonical forms of functions of two letters of even degrees, and possibly 
to exhibit some important applications of the principles of the method to the 
theory of numbers. 



APPENDIX. 

Note (A). 

The permutants (meaning, in Mr Cayley's language, the hyperdeter- 
minants) of -Fai+i {x, y) of the fourth dimension in respect to the coefficients 
of F, may be all obtained by taking the quadratic permutant in respect to x 
and y of the quadratic permutant in respect of ^ and r; of 

I having any integer value from 1 to n. 

In extension of a theorem in the foregoing Supplement, which applies 
only to the case of Z = n, I am able to state the following more general 
theorem, in which the same notation is preserved as above [page 207]. 
The quadratic permutant in respect to f and r) of 



(2n + l)2n...{2n-2l + 2) V dx ' dy 
is equal to 

If now we proceed to form the quadratic permutant of the above sum 
in respect to x and y, we know d, priori, by reason of Mr Cayley's invaluable 
researches, that we shall not get radically distinct results for all values, but 
only for certain periodically changing values of I. 



34] On Canonical Forms. 215 

I have not yet had leisure to seek for an explicit demonstration of this 
remarkable law, founded upon the above given canonical representation. 

Note (B). 

The lemma, upon which the general method for reducing odd degreed 
functions to their canonical form is founded, may be stated rather more simply 
and more generally as follows : — 

The determinant 

T T T 



where Tg denotes Aj^a^^ + A2a2^ + ... + A^am^ provided that m is less than n, 
is identically zero. In the theorem, as thus stated, there is no substantial 
loss of generality arising from the omission of the b's. 

Thus stated the theorem and its extensions evidently repose upon the 
same or the like basis as the theory of partial fractions. 

Note (0), referring to the original " Sketch." 

The Boolo-Hessian scale of determinants furnishes a very pretty general 
theorem of geometrical reciprocity in connexion with the doctrine of suc- 
cessive polars. Let F(x, y, z), a cubic homogeneous function of x, y, z 
equated to zero, "express in general a curve of the third degree ; then 

T- + 6-j-+CT-)i'' will express its first polar in respect to the point a, b, c, 

that is, the conic which passes through the six points in which the tangents 
drawn from a, b, c to touch the given curve meet the same. 
Again, if we take I, m, n the coordinates of any new point, 
, d d d\ / d J d , d\ „ 

dx dy dzJ \ dx dy dzJ 

will express the polar, that is the chord of contact of the above conic, in 
respect to the last named point. If now we eliminate I, m, n between the 
three equations 

J J J ^ / J J ^ \ 

F=0, 
F = 0, 
F=0, 



, c? , d ^ d\ f 
dx dy dzJ \ 


— + b 
dx 


d 

dy 


d' 
dz, 


J d , d , d\f , 
dx dy dzJ \ 


d ^, 
dx 


d 
dy 


d\ 
dz) 


, d d d\ / , 
, dx dy dzJ \ 


d ^, 
dx 


d , , 
dy 


d' 
dzj 



216 On Canonical Forms. [34 

it is easily seen that the resultant of the elimination is the square of the 
determinant 



a', V, I 

a", h", I 

multiplied by the Hessian of the given function. And, moreover, that if we 
eliminate x, y, z we shall obtain precisely the same result with the letters 
I, m, n substituted for x, y, z. Hence it follows, that if we take the doubly 
infinite system of first polars to a given curve of the third degree, in respect 
to all the points lying in its plane, and then from any point in the Hessian 
to the given curve, draw pairs of tangents to each conic of the system so 
generated, then all the chords of contact will meet in one and the same point, 
which will itself be also a point situated upon the Hessian and conjugate to 
the former. 

So, in general, for a function of any degree of any number of letters, 
viewed with relation to the doctrine of successive polars, the determinants 
of the Boolo-Hessian scale take one another up in pairs ; namely the first 
takes up the last but one, the second the last but two, and so on ; and 
consequently, if the degree of the function be odd, that function which 
(making abstraction of the constant determinant at the end) lies in the 
middle of the scale pairs with itself, and, in a sense analogous to that above 
exhibited for a function of the third degree, may be said to be always 
its own reciprocal. 

P.S. I have just discovered the method of reducing functions of two 
letters of even degrees to their canonical form, which will shortly be published 
in a second Supplement. 

At present I offer the annexed theorem (which strikingly contrasts with 
the law of uniqueness demonstrated of functions of an odd degree) as a 
foretaste of the enchanting developments with which I hope shortly to present 
my readers :— 

If a given homogeneous function of x and y of the degree 2n he supposed to 
he thrown under its canonical form, 

U-c'^ + M2=" + . . . + M„2" ■]- K {U^U,, . . . Unf, 

then will K'^ have n^ — 1 in general distinct values, to each of which will 
correspond a single distinct system of the linear functions of x and y, 
11 1 

1"Mi, 1"m2, ... 1"m„. 



35. 



EXPLANATION OF THE COINCIDENCE OF A THEOREM GIVEN 
BY Mr SYLVESTER IN THE DECEMBER NUMBER OF THIS 
JOURNAL, WITH ONE STATED BY PROFESSOR DONKIN 
IN THE JUNE NUMBER OF THE SAME. 

[Philosophical Magazine, (Fourth Series) i. (1851), pp. 44—46.] 

I WISH to state, without loss of time, that in the theorem given by me * for 
the composition of two successive rotations about different axes, I have been 
anticipated by Prof. Donkin in the June Number of your Journal. 

To my shame I must confess, that, although an occasional contributor to, 
I am not invariably a constant reader of your valuable miscellany, otherwise 
I should not have introduced the theorem in question without due acknow- 
ledgment of Professor Donkin's claims to whatever merit may attach to the 
priority of publication. The fact is, that I made out the theorem for myself 
nine years ago, and had some communication on the subject with Professor 
De Morgan, who was then writing the seventeenth chapter of his Differential 
Calculus. A recent conversation with this gentleman has brought back to 
my mind a vivid recollection of the course of that communication. I brought 
under Professor De Morgan's notice the analytical memoir of Sr Gabrio Pola 
on the subject in the Memoirs of the Italian Society of Modena, and satisfied 
myself of the existence of the single axis of displacement by compounding 
the two rotations in the manner given in my paper, which, for the case of two 
axes fixed in space, is the same as Professor Donkin's, and for two axes fixed 
in the rotating body is materially, although not formally the same. 

It then occurred to me that a more simple demonstration ought to be 
deducible from the possibility of always finding the point on a sphere, by 
revolution about which, as a pole, one equal arc could actually be shown to 
be transportable into the place of another. But in proceeding to work out 
this idea I fell into a remarkable blunder, in which I have since been followed 
by more than one able friend to whom I have proposed the question. The 
[* p. 158 above.] 



218 Explanation, etc. [35 

blunder was of this kind: — Two arcs have to be drawn, bisecting at right 
angles the arcs joining the extremities of two equal arcs; the point of inter- 
section of the two bisecting arcs viust in all cases fall outside the quadrilateral 
formed by the equal and joining arcs. I supposed it to fall inside. There 
appears to be a fatal tendency to do so in all who take the subject in hand. 
In consequence of this error, the cause of which I did not at the moment 
perceive, I was driven to deny and admit in one breath the same proposition. 
Mr De Morgan sent me the correct proof after this method (the same as that 
given by him at page 489 of his Calculus), I am inclined to think after I had 
myself detected my error ; but of this I cannot feel certain. 

This is the method alluded to by me in the words " it is right to bear in 
mind, &c.," at the time of writing which all recollection of the same thing 
having been published by Mr De Morgan had vanished from my memory. 

The proof of the triangle of rotations is so simple, that, as Professor 
Donkin states (in a letter which he has done me the favour of addressing me 
on the subject) was the case with himself, I thought it incredible that it 
should not have appeared in some elementary work, and I was therefore at 
no pains to publish it as my own ; nor should I have written at all on the 
subject, had it not been for the surprise occasioned to my mind by falling in 
with Professor Stokes's article in the Cambridge and Dublin Mathematical 
Journal, to demonstrate the existence of an instantaneous axis, which 
proceeds in apparent unconsciousness of the so simply demonstrable law, 
that any number of rotations of any kind (and therefore those that take 
place in an instant of time) are representable by a single rotation about 
a single axis. I shall feel obliged by the early insertion of this explanation, 
more in justice to myself than to Professor Donkin, whose high and worthily 
earned reputation, not to speak of the disinterested love of truth for its own 
sake, apart from personal considerations, which animates the labours of the 
genuine votary of science, must make him indifferent to whatever credit 
might be supposed to result from the first authorship or publication of the 
very simple (however important) theorem in question. 



36. 



AN ENUMERATION OF THE CONTACTS OF LINES AND 
SURFACES OF THE SECOND ORDER. 

{Philosophical Magazine, i. (1851), pp. 119 — 140.] 

It is well known that in general any two homogeneous quadratic 
functions of the same system of variables may be simultaneously trans- 
formed, so as to be expressed each of them as pure quadratic functions of 
a new system of variables equal in number and linearly connected with the 
original ones; a pure quadratic function meaning one in which only the 
squares of the variables are retained. 

Every homogeneous quadratic function may be treated as the character- 
istic* of a locus of, the second degree : if the function be of two letters, the 
locus is a binary system of points in a line wherein the distances of two 
fixed points from either point of the given system or given multiples of such 
distances correspond to the variables ; if of three letters, the locus is a conic, 
the distances or given multiples of the distances of every point in which from 
three given lines in the plane of the conic are represented by the variables; 
if of four letters, the locus is a surface of the second order, the coordinates 
being the distances or multiples of the distances of any point therein from 
four planes drawn in the space in which the surface is contained, and so on 
for loci of four and higher dimensions. 

I propose, however, in the present paper to restrict myself to the theory 
of the contacts of loci not transcending the limits of vulgar space, by which 
I mean the space cognizable through the senses f, and shall accordingly be 

* According to the definition stated by me in a previous paper, the characteristic of a locus is 
the function which, equated to zero, constitutes the equation thereto. 

t If the impressions of outward objects came only through the sight, and there were no sense 
of touch or resistance, would not space of three dimensions have been physically inconceivable ? 
The geometry of three dimensions in ordinary parlance would then have been called trans- 
cendental. But in very truth the distinction is vain and futile. Geometry, to be properly 
understood, must be studied under a universal point of view ; every (even the most elementary) 
proposition must be regarded as a fact, and but as a single specimen of an infinite series of 
homologous facts. 

In this way only (discarding as but the transient outward form of a limited portion of an 
infinite system of ideas, all notion of extension as essential to the conception of geometry, 
however useful as a suggestive element) we may hope to see accomplished an organic and vital 
development of the science. 



220 An Enumeration of the Contacts of Lines [36 

almost exclusively concerned in determining the singular cases of conjugate 
systems of quadratic forms of two, three, and four letters respectively. 

In order that the reduction of any such system, say U and V, to a pure 
quadratic form may be possible (as it generally is), it is necessary that none 
of the roots of the complete determinant of C/^ + X F shall be equal ; if any 
relation of equality exist between these roots, tlie general reduction is 
generally no longer possible ; under peculiar conditions, however, as will 
hereafter appear, in spite of the equality of certain of the roots, the 
irreducibility in its turn will cease, and the ordinary reduction be capable 
of being effected. It is easily seen, that to every relation of equality between 
the roots of the determinant of U + W must correspond a particular species 
of contact between the loci which U and V characterize. But we should 
make a great mistake were we to suppose that every such relation of equality 
corresponded with but one species of contact ; for instance, the characteristics 
of [/" and Fof two conies are functions of three letters, and d{U + W) will 
be a cubic function of X,. Such a function may have two roots, or all its roots 
equal : this would seem to give but two species of contact, whereas we well 
know that there are no less than four species of contact possible between two 
conies. Accordingly we shall find, that, in order to determine the distinctive 
characters of each species of contact, we must look beyond the complete 
determinant, and examine into the relations (in themselves and to one 
another) of the several systems of minor determinants that can be formed 
from JJ + XV. 

By pursuing this method, we may assign a priori all the possible species 
of contact between any two loci of the second degree. How important this 
method is will be apparent from the fact, that not only have the distinctive 
characters of the various contacts possible between surfaces of the second 
order never been determined, but their number and the nature of certain of 
them have remained until this hour unknown and imsuspected. 

The method which we shall pursue is an exhaustive one, and will conduct 
lis by a natural order to a systematic arrangement of all the different modes 
and gradations of such contacts. 

In a paper* in this Magazine for November 1850, 1 explained the decline 
of minor determinants, and stated a law, called the homaloidal law, con- 
cerning them. 

If U and V be characteristics of the two loci whose contacts are to be 
considered, U+XV will be the function, the properties of whose complete 
determinant, and of the minor systems of determinants belonging to it, will 
serve to specify the nature of the contact. 

It will be remembered, that, whatever be the number of variable letters 
in any quadratic function U, three of its first minor determinants being zero, 
[* p. 150 above.] 



36] and Surfaces of the Second Order. 221 

makes all the first minors zero ; six of its second minors being zero, makes all 
the second minors zero ; and so on for the third, fourth, &c. minor systems 
according to the progression of the triangular numbers. 

It is well known that whatever linear transformations be applied to 
a quadratic function W, the complete determinant thereof will remain un- 
altered, except by a multiplier depending upon the coefficients introduced 
into the equations of transformation; consequently the roots of \ in the 
equation obtained by making the determinant of V+XV zero remain 
unaffected by such transformation ; and any relation or relations of equality 
among the roots of the equation a{U+\V) = Q is an immutable property 
of the system U, V, which is unaffected by linear transformations. Another 
and more general kind of immutable property (comprehending the above as 
a particular case), to which I shall have occasion to refer, is the following. 

Suppose all the minors of any order of U + XV have a factor X + e in 
common ; this factor will continue common to the same system of minors 
when U and V are simultaneously transformed. This is a very important 
proposition, and easily demonstrated ; for if \ + e be a common factor to all 
the rth minors of U + \V, (U - eV) will have its rth minors zero, and there- 
fore, as explained by me in the paper above referred to, U— eV will be 
degraded r orders below U or V. This is clearly a property independent 
of linear transformation, consequently X + e will remain a factor of the 
transformed rth minors. 

In like manner it is demonstrable that any number of distinct factors 
X + ei, X+e^... common to the rth minors of one form of ?7-l-XF, will 
remain common factors of any other linearly derived form of the same. 
It is consequently necessary that each rth minor of one form of any 
quadratic function W shall be a syzygetic* function of all the rth minors 
of any other form of the same ; and consequently a function of X of any 
degree, whether all its factors be or be not distinct, which is common to 
the rth minors of one form of U + XV, will remain so to the rth minors of 
any other form of the same. 

The law exhibiting the connexion of each rth minor of one form of W 
(any homogeneous quadratic function) with all the rth minors of any other 
form of W, will form the subject of a distinct communication. 

Finally, to fully comprehend the annexed discussion, the following 
principle must be apprehended. 

* If A = pL + qM+rN + &c., where p, q, r ... do not any of them become infinite when 
L, M, N ... or any of them become zero, A may be termed a syzygetic function of i, M, N.... 

In the theorem above alluded to, it will be shown (as might be expected) that the syzygy in the 
case concerned is of the simplest kind, that is, that each rth. minor of a quadratic function of any 
number of letters is a homogeneous linear function of all the j-th minors of the same quadratic 
function linearly transformed. 



222 An Enumeration of the Contacts of Lines [36 

If any factor K^ enter into all the ?'th minors of W, and if ^* be the 
highest power of K common to all the (r + l)th minors, then K'^"'- will be 
a common factor to all the (r — l)th minors. 

Let r be taken unity; it is easily proved* that the complete determinant 
of any square matrix may be expressed by the difference between two pro- 
ducts+, each of two first minor determinants divided by a certain second 
minor determinant. The proposition is therefore demonstrated for this case, 
and thereby in fact implicitly for every case, inasmuch as the first minors of 
every rth minor are (r+l)th minors of the original matrix. Hence it 
follows, that if any system of rth minor determinants have a common factor 
6*, the complete determinant must contain at lowest the factor e^'+^i*, and any 
sj'stem of (r — s)th minor determinants thereunto will contain at lowest the 
factor e<«+^ii 

I now proceed to apply these principles to the determination of the 
relative forms of conjugate quadratic functions representing geometrical 
loci of the second order. I shall begin with two binary systems of points 
in a right line. 

The general characteristics U and V of two such systems may be thrown 
under the form 

U = x\+y'' 1 

V=ax" + by^)' 

When D (V+\U) = has its two roots equal, these systems have a point 
in common. The above forms cease to be applicable, and convert into 

U=xy y 

V = ax^ + bxy ) 
where x = represents the common point. 

* This will appear in my promised paper on Determinants and Quadratic Functions. 

t When the matrix is symmetrical about one of its diagonals (as it is in the case which we are 

concerned with), one of these products becomes a square. I may take this occasion of hinting, 

that the theory of quadratic functions merges in a larger theory of binary functions, consisting of 

the sum of the multiples of binary products formed by combining each of one set of quantities, 

X, y, z ... with each of the same number of quantities of another set, as x', y', z' .... For 

instance, 

axx' + bxy' + cxz' 

+ a'yx' + h'yy' + c'yz' 

+ a"zx' + b"zy' + c"zz' 

would be a binary function, and its determinant (no longer, as in a quadratic function, 

symmetrical about either diagonal) would correspond to the square matrix 

a 1) c 

a' h' c' 

a" h" c". 

Almost all the properties of quadi'atic apply, with slight modifications, to binary functions. 



36] and Surfaces of the Second Order. 223 

iLet U and V now represent two conies. When there is no contact, we 
have as the types of their characteristics 

[7 = a^ 4- 2/- + 0^, 

V=ax" + by'' + cz"^. 

The three roots of n (F+ XD') = are 

X = — a, \ = — b, \ = — c, 

showing that there are three distinct pairs of lines in which the intersections 
of U and V are contained, the equations to three pairs being respectively 

{a — c)os- + (b — c)y- = 0; 

the four points of the intersection being defined by the equations corre- 
sponding to the proportions 

x:y:z:: >J{b — c) : \/(c — a) : ^/(a, — b). 

Now let a(U+XV) have two equal roots ; the characteristics assume the 
form 

TJ = x- + y"- + xz, 

V = ax- + by- + cxz *. 

Two of the pairs of lines become identical, that is, two of the four points of 
intersection coincide. 

* We may if we please make a = i; for it may be shown that the equations, in their present 
forms, contain an arbitrariness of 10 degrees; namely, 9 on account of x, y, z being arbitrary 
linears of f, i;, S ; 2 on account of the ratios a:h:c\ together 11 reduced by one degree on 
account of a;, xj, z, changed into Ix, ly, h, leaving f7=0, V=0 unaffected. Now the degrees of 
arbitrariness in two conies, subject to satisfy only one condition, is 2 x 5 - 1 or 9. Hence there 
is one degree of arbitrariness to spare. In fact, on mating a =6, the axis z becomes the line 
joining the two points of intersection distinct from the point of contact ; x remaining the tangent 
at the point of contact, and y, strange to say, still arbitrary, subject only to passing through the 
point of contact ; if, however, y be made to pass through the point of contact, and either one of 
the distinct intersections, this form, 

U=x- + y^ + xz, 

V=ax- + ay^ + cxz, 
becomes no longer tenable, but gives place to 

U=y''+yx+xz, 

F= ay^ + ayx + cxz, 

where x is the tangent at the point of contact, z the Kne joining the two intersections with one 
another, and x, x + y respectively the lines joining either of them with the point of contact ; if the 
multiplier of yx in V in the above be made b instead of a, x remains the tangent as before, y 
becomes any Une through the point of contact, and z any line through one of the distinct inter- 
sections. A systematic view of similar modulations of form and the study of the laws of 
arbitrariness connected with them, as applicable to the general subject-matter of this paper, 
must be -deferred to a subsequent occasion. 





(1) 




(2) 


z = 0, 


(3) 


2 = 0. " 


(4) 



224 An Enumeration of the Contacts of Lines [36 

This may be termed " Simple Contact." The tangent at the point of 
contact is x = 0; this equation making JJ and V each become of only one 
order. 

The intersections are 

x = 0, 2/ = 0, 

x = 0, y = 0, 

V(» — c) « + \/(6 — c) 2/ = 0, 

^/(a — c)x — \/Q) — c)y = 0, 

These are obtained by making V—aU=0, which gives a; = or ^^ = 0. 

x = gives y^ = 0, that is, y = twice over, and = gives 

(a-c)x' + {b-c)y^ = 0. 

The number of conditions to be satisfied in this case is one only. 

Next let n{V'+XV) have all its roots equal. This condition will be 
satisfied (still leaving U and V as general as they can remain consistent 
with these conditions) by making 

U = X- + yz + yx, 

y = ax"^ + ayz + hyx. 

Here only one distinct pair of lines can be drawn to contain the inter- 
sections, showing that three out of the four points come together. 

This may be termed "Proximal Contact." The number of affirmative 
conditions to be satisfied is two, and the contact is therefore entitled of the 
second degree. 

The tangent at the point of contact is y = 0, and the four intersections 
become 

x = 0, y = 0, 

a; = 0, y = 0, 
x=0, y = 0, 
x = 0, z = 0. 

These may be obtained from the equation V—aU=0, which gives y = 
or 2 = 0; the former implying concurrently with itself x^ = 0, and the latter 
yz=0. 

Thus we obtain three systems, 

a; = 0, J/ = 0, 
and one 

x = 0, z = 0, 

corresponding to three consecutive points and the single distinct one. 



36] and Surfaces of the Second Order. 225 

The determinant of U +\V being only of the third degree in X, we 
have exhausted the singularities of the system JJ, V dependent on the form 
of the complete determinant of U+XV. 

Let now the first minors of JJ + XV have a factor in common; this will 
indicate that U+XV may be made to lose two orders by rightly assigning X, 
in other words, that the intersections of U and V are contained upon a pair 
of coincident lines. Here it is remarkable that the original forms of IT and V 
reappear, but with a special relation of equality between the coefficients : we 
shall have, in fact, 

U = x^ \- y°' + z", 
V = ax^ + ay" + bz'^. 
This gives the law for double, or, as I prefer to call it, diploidal contact*. 
By virtue of the Homaloidal law, we know that if three first minors of 
U +XV be zero, all are zero; we have therefore to express that three 
quadratic functions of \ have a root in common. This implies the exist- 
ence of two affirmative conditions ; the contact of the two conies taken 
collectively may therefore be still entitled of the second degree, although 
the contact at each of the two points where it takes place is simple, or 
of the first degree. 

These points are evidently defined by the equation 

[x■\-^J{-\)y = 0, = 0}, 

{*-V(-l)2/ = 0, = 0}, 
and the ordinary algebraical' solution of the equations U=0, V=0 would 
naturally lead to the four systems 

« + V(- 1) 2/ = 0, = 0, 

x+^{-l)y=0, = 0, 

a; - V(- 1) 2/ = 0, = 0, 

«-V(-l)2/ = 0, 0=0; 

the two tangents at the point of contact are x + \/(— 1) y = 0, x — \/(— 1) 2/ = 0, 
and the coincident pair of lines containing the intersections is z^ = 0. 

* See my remarkst on the conditions which express double contact in the Cambridge Journal, 
. Nov. 1850. If n functions, being all zero, be the condition of a fact, but r independent syzygetie 
equations admit of being formed between these functions, the number of affirmative conditions 
required is not n, but (« - r) ; because the fact may be expressed by affirming (ra - r) equations and 
denying certain others. Thus if P=0, Q = 0, R = 0, S=0 express a fact, and 
PP' + QQ' + RR' + SS' =0, 
PP" + QQ" + RR" + SS"=0, 
the fact is expressible by alBrming P = 0, Q = 0, and denying R'S" -R"S'=0, for then P=0, Q=0 
will imply R = 0, S=0; or, in like manner, by affirming any other two out of the four necessary 
equations, and denying the other equations. Observe, however, that all the required equations 
viay coexist in the absence of such right of denial. 
[t p. 129 above.] 
s. . 15 



226 An Enumeration of the Contacts of Lines [36 

It may at first view appear strange, that whilst no condition is required 
in order that U and V may be simultaneously metamorphosed into the forms 
of a;- + y^ + ^^ ax^ -\- by'^ -\- cz^ , a, h and c being all unequal, for this metamor- 
phosis to be possible when any two become equal, not one but two conditions 
must be satisfied. The reason of this is, that the coefficients of transform- 
ation, which, as well as a, h, c, are functions of the coefficients of the given 
quadratic functions, become infinite oq constituting between the said 
coefficients such relations as are necessary for satisfying the equation a = h, 
or a = c, or h = c, except upon the assumption of some further particular 
relations between them over and above that implied in such equality. 

In the ordinary case of diploidal contact, the first minors having a factor 
in common, this factor will enter twice into the complete determinant of 
U + XV, but it inay enter three times: this will indicate, that not only 
do the four intersections lie on a coincident pair of lines, but furthermore, 
that there is but one pair of lines of any kind on which they lie. 

In the ordinary case of diploidal contact, it will be observed that this 
latter condition does not obtain ; the four intersections lie on a coincident 
pair of lines ; but they lie also on a crossing pair, namely, in the two tangents 
at the points of contact. In this higher species of diploidal contact, it is 
clear that the two points of contact, which are ordinarily distinct, come 
together, and that all four intersections coincide. 

This I call confluent contact ; the forms of U and V corresponding thereto 
will be 

U = X'+'>f + xz, 

V = ay^ + axz ; 

the common tangent at the point of contact being a; = 0, and the four 
coincident points oe' = 0, y"^ = 0. 

The number of affirmative conditions to be satisfied being three, the 
contact is to be entitled of the third degree. 

Observe, that it is of no use to descend below the first minors in this 
case ; because the second minors, being linear functions of X, could not have 
a factor in common, unless V : U becomes a numerical ratio, which would 
imply that the conies coincided*. 

Fortified by the successful application of our general principles to the 
preceding more familiar cases of contact, we are now in a condition to apply 
with greater confidence the same a priori method to the exhaustion and 
characterization of all the varied species of contact possible between surfaces 

* No-contact and complete coincidence may be conceived as the two extreme oases in the scale 
of relative conjugate forms. 



36] and Surfaces of the Second Order. 227 

of the second order ; a portion of the subject comparatively unexplored, and 
never before thought susceptible of reduction to a systematic arrangement. 

When there is no contact, we may write 

U = x" + y'' + z" + f", 

V=ax^ + htf + cz" + dt^, 

and the intersection of the surfaces will lie in each of the four cones, 

{a-d)x^-\-{h-d)y'-\-{c -d).0^ = O, 

(a - &) «^ + (c - h) z' + id- h) t' = 0, 

(a - c)x^ + (b- c) y' + {d- c) P = 0, 

{b -a)xf + (c- a) z- + {d- a) f = 0. 

Whenever the surfaces are in contact, certain of these cones will coincide 
with certain others, so that their number will be always less than four. Also, 
as we shall find in such event, they may degenerate into pairs of intersecting 
or coincident planes. 

Let us begin with considering the cases of contact for which the first 
minors (and consequently d fortiori the minors inferior to the first) have 
no factor in common. 

Here a {V+XV) is a biquadratic function. 

If X have all its roots unequal, we have U and V as above given. 

If two roots are equal, the characteristics assume the form 

U = x^ + y- + z'^ + xt 

V = ax^ + by'- + cz~ + dxt j 

The touching plane is x = 0; the point of contact is x = 0, y=0, z = Q; the 
curve of intersection is one of the fourth degree, with a double point at the 
point of contact. 

There is but one condition to be satisfied, and the contact may be entitled 
" simple " and of the first degree. 

Next let X have three equal values, the equations become 

U=x^ + yz + f + xy, 

y = x^ + yz + at- + bxy. 

The tangent plane at the point of contact y = 0, and the point itself x = 0, 
y = 0, t = 0. The curve of intersection is a curve of the fourth ordei-, with a 
cusp at the point of contact. The number of affirmative conditions to be 
satisfied is two ; the contact is of the second degree, and may be termed 
" proximal " or cuspidal. 

15—2 



228 An Enumeration of the Contacts of Lines [36 

Next let D ( ?7 + \ F) have two pairs of equal roots, we shall find 
U =x'^ + xy + zt, 

V = ayz + hxy + czt. 

The line x = Q, z = will be common to both surfaces. The curve of 
intersection will therefore break up into a right line and a line of the 
third order. 

The former will meet the latter in two points, which will be each of them 
points of contact. The contact is therefore diploidal ; but as there is another 
species of diploidal contact to which we shall presently come, it will be 
expedient to characterize each of them by the nature of the intersections 
of the two surfaces ; accordingly this may be termed unilinear-intersection 
contact, or more briefly, unilinear contact. 

The number of affirmative conditions to be satisfied being two, it may 
be said to be collectively of the second degree, but (obviously ?) the contact 
at each of the two points is of the nature of simple contact. 

Lastly, let us suppose that all four roots of U + XV are equal; we shall 
find, as the most simple expressions of the most general forms of the two 
surfaces, 

U = x^ + xy + yz + zt, 

V = axy + bz" + azt. 

In this case the two points of intersection of the curve of the third 
degree, and the right line on which the surfaces intersect, come together, so- 
that the right line becomes a tangent to the curve. The number of conditions 
to be satisfied is three : there is but one point of contact which may be con- 
sidered as the union of two which have coalesced, and the species may be 
defined as confluent-unilinear contact. 

If we throw the equations to the conoids having an unilinear contact into 
the form 

x(x + y) + zt = 0, 

xy + z(y + ct) = 0, 
we obtain 

(x + y){y + ct)-yt = 0, 

which last equation is no longer satisfied by x = 0, z = 0, these systems of 
roots having been made to disappear by the process of elimination. 

The curve of the third degree, in which the two given conoids intersect^ 
may thus be defined as their common intersection with the new conical 
surface defined by the third of the above equations. 



36] and Surfaces of the Second Order. 229 

More generally, it is apparent that the three conoids, 
xu — yt = 0\ 
yv — zu = Q\ , 
zt — XV = ()] 

in which x, y, z, t, u, v may any of them be considered aa a homogeneous 
linear function of four others, intersect in the same line of the third degree. 
Besides which, the first and second intersect in the right line y, u ; the second 
and third in z, v; the third and first in x, t; each of which lines it is evident 
is a chord of the common curve of intersection. For instance, y = 0, v = 
may be satisfied concurrently with all the above three equations by satisfying 
the equation zt — xv = 0, which, as two linear relations exist originally be- 
tween the six letters, and two more have been thrown in, becomes a quadratic 
equation between any two of the letters. 

The only case of exception to this reasoning is, when y = 0, ^^ = can be 
satisfied concurrently with z = 0,v = Q, and with « = 0, i5 = ; but in this case 
the surfaces all become cones ; and as there is no longer a curve of the third 
degree, " Cadit queestio." Even here, however, the intersection of any two of 
the surfaces becomes a conic, and two coincident generating lines on the two 
cones ; so that if we take one of these and the conic to represent a degenerate 
form of a line of the third degree, the remaining straight line passes through 
a double point of such degenerate form, and the case passes into that of 
' confluent-unilinear contact. 

The two double points in the intersection of the two conoids 

JJ = x{x + y) + zt = 0, 

V=xy + z{y-\-ct) = 0, 

by which I mean the points of intersection of the conic with the right line 
common to them, are found by making x = 0, z = Q, and substituting in the 
derived equation 

{x ■{-y){y + ct) -ty = 0, 

which gives y = 0,ory+{c—l)t = i); so that the two points required are 

a; = 0, 2/ = 0, z = 0, 

x = 0, y = {l-c)t, = 0. 

It appears also that the entire intersection is contained in each of the two 
cones, 

U-V, that is, x''-{-z{(\-c)t-y] 
and 

cU—V, that is, cx^ + y [{c-l)x — z], 

the respective vertices of which are at the points above determined. 



230 A7i Enumeration of the Contacts of Lines [36 

The equations for confluent-unilinear contact, 
x{x + y) + z{y + t) = Q, 

xy + 2(cz + t) = 0, 
give 

{x+y)(cz + t)-(y + t)y = 0] 

which, on making x = 0, 2 = 0, is satisfied by y^ = 0; showing that the 
confluence takes place at the point 

x = 0, y = 0, z = 0. 

The number of terms in the two equations for ordinary unilinear contact 
being six, and in those given for confluent unilinears seven, and the empirical 
rule in all other cases being that the terms tend to diminish and never 
increase in number as the degree of the contact (expressed by the number 
of conditions to be satisfied) rises, I am led to suspect that the conjugate 
system for the latter species of contact may admit of being reduced to some 
more simple form. 

I must state here once for all, that all the distinct systems of (at least 
consecutive) conjugate forms that have been, and will be given, are mutually 
untransformable. This it is which distinguishes singular from particular 
forms. 

A particular form is included in its primitive ; but a singular form is one, 
which, while it responds to the same conditions as some other more general 
form, is incapable of being expressed as a particular case of the latter, on 
account of the additional condition or conditions which attach to it. 

I pass now to the singularities which arise from the first minor deter- 
minants of U+XV having a factor in common, the second minors being 
supposed to be still without a common factor. 

When this common factor is linear in respect to X, let it be supposed 
to enter not more than twice (twice, we know, by the general principle 
enunciated at the commencement of this paper, it must enter) into the 
complete determinant. 

Two of the cones containing the intersection of U and Y then become 
coincident, and degenerate each into the same pair of crossing planes. This 
may be termed biplanar-contact. The characteristics of such contact are 

U=x" + y'' + z- + t'', 

V = ax^ + ay'^ + bz^ + ct" ; 

the points of contact are two in number, being at the intersection of the two 
plane conies into which the curve of intersection breaks up. The two planes 



36] and Surfaces of the Second Order. 231 

in which these lie are given by the equation (6 — a) z^ + {c — a)t=Q; these 
intersect in the right line z = 0, t = Q, which meets both surfaces in the same 
two points, 

z = 0, t = 0, x+>J(-l)y = 0, 

z=0, t = 0, a;- V(-l)2/ = 0, 

the two common tangent planes at these points being 

«; + V(-l)2/ = 0, fl;-V(-l)2/ = 
respectively. 

This, then, is another species of double contact between two conoids, and, 
as far as I know, the only kind hitherto recognized as such. The number of 
conditions to be satisfied remains two, as in the former species. 

Next suppose that the common factor of the first minor enters three 
times into the complete determinant instead of huice only, as in the last 
case. 

The corresponding characteristics will be found to be 

U=x- + zt + 'tf + z^, 

V = ax' + azt + by" + cz-. 

The intersection of U, V still lies in two planes, 

{b-a)f + (c-a)2^ = 0; 

but the intersection of these two planes, 

y = 0, z = 0, 

meets the surfaces in the two coincident points, 

2/ = 0, z = 0, a;' = 0. 

This, therefore, I call confluent-biplanar contact; the two conies con- 
stituting the complete intersection, instead of cutting, touch and at their 
point of contact the two conoids have a contact of a superior order. The 
conditions to be satisfied for this case are three in number. 

Next suppose that the common factor of the first minors enters only 
twice into the complete determinant, but that the remaining two factors 
become equal. 

Here the analytical characters of unilinear and biplanar contact are 
blended ; in fact, the intersection consists of a conic and a paii- of right 
lines meeting one another and the conic. The characteristics are 

U = a^ + y' + z'' + zt, 

V=aiif+ ay^ + hz^ + czt. 



232 An Enumeration of the Contacts of Lines [36 

The intersection is contained in the two planes 

z = 0, {h — a)z + {c-a)t = Q, 

and consists of the two lines z = 0, x' + y'' = 0, lying in the common tangent 
plane z = 0, and the conic 

{b-a)z+ {c-a)t = 

(a — c)cc- + (a — c) y^- + (b — c)z'^ = \ 

There are three points of contact, namely, the point x = 0, y = 0, z=0, 
where the two right lines cut, and a;^ + y^ = 0, t = 0, z = 0, where these lines 
meet the conic. This, then, is a case of triple contact. I distinguish it by 
the name of bilinear-contact. The number of conditions is still three. 

Now all else remaining as before, let the two pairs of equal roots in the 
complete determinant become identical, or, in other words, let the common 
factor of the first minors be contained four times in the complete deter- 
minant. The characteristics become 

U = xz + cct + y^ + z^, 

V = axz + hxt + hy'' + hz''. 

The intersection becomes the two right lines 

a; = 0, y- + z'^ = 0, 
and the conic 

z = 0, «- + 2/^ = 0. 

All these meet in the same point, 

a; = 0, 2/ = 0, = 0; 

so that instead of contact in three points, the contact takes place about one 
only, in which the three may be conceived as merging. This I call confluent- 
bilinear contact. It requires the satisfaction of four conditions. 

Next let us suppose that the two distinct factors are common to each 
of the first minors. This will imply the existence of four affirmative 
conditions. 

The complete determinant will of necessity contain each of these factors 
twice, so that no additional singularity can enter through this determinant. 
The characteristics assume the form 

'U = x'' + y^ + z^ + t\ 

V=aa^ + ay^ + bz^ + bt": 

The two surfaces will meet in four straight lines, forming a wry quadrilateral, 
whose equations are 

« + V(-l)2/ = 0. 

z±'^{-l)t =0. 



36] and Surfaces of the Second Order. 233 

These intersect each other in the four points 

a; = 0, 2/ = 0, Z- + P =0, 

z = 0, t = 0, «2 + 2/= = 0, 
each of which will be a distinct point. This I term quadrilinear contact. 

Now let the two factors common to each of the first minors become 
identical ; so that a squared function, instead of an ordinary quadratic 
function of X, is now their common measure. 

The factor which enters twice into each of the first minors will enter 
four times into the complete determinant; the number of conditions to be 
satisfied is , one more than in the preceding case, namely five, and the 
characteristics become 

U = x'^ + y^ + a;2 + yt, 

V = ace- + by- + cxz + cyt. 

Here arises a singularity of form in the intersections utterly unlike 
anything which has been remarked in the preceding cases. For it will 
not fail to have been observed, that the intersection in the nine preceding 
cases was always a line or system of lines of the fourth degree, so as to be cut 
by any plane in four points. 

But in this case, the fact of the first minors having a factor in common, 
shows that the intersection is contained in two planes (which is of course to 
be viewed as a degenerate species of cone) ; and the fact of the complete 
determinant having all its roots equal, shows that there is but one system of 
a pair of planes in which the intersection is contained, and no more. 

So that the two pairs of planes, into which the wry quadrilateral was 
divisible in the case immediately preceding, now become a single pair. This 
can only be explained by two of the opposite sides of the quadrilateral 
becoming indefinitely near to one another, but still not coinciding in the 
same planes; so that the actual visible or quasi-visible* intersection will 
be in three right lines, of which the middle one meets each of the 
two others. 

This will further appear by proceeding regularly to solve the equations 
?7=0, F=0. 

V— cU=Q gives y — ± kx, where ^" = a / ( 7 ) > ^^^ therefore xz + kxt = 0, 

or xz — kxt = ; whence we see that the complete intersection is represented 
by the lines 

(a; = 0, 2/ = 0) ; {z + kt = 0, y-hx = 0), 

(» = 0, 2/ = 0) ; {z-kt=0, y + kx = 0), 

* I use the term quasi-visible, because the intersection may become in part or whole 
imaginary. 



234 An Enumeration of the Contacts of Lines [36 

showing that there are but three physically distinct lines, as already 
premised. 

This, then, may be considered as derived from the preceding case of 
a wry quadrilateral intersection, by conceiving two opposite sides of the 
quadrilateral to come indefinitely near, but without coinciding. 

Let these two lines be called P and P' ; take any point in P and any two 
points in P' indefinitely near to one another and the point first taken, then 
this indefinitely small plane will be common to both surfaces, and consequently 
they ought to touch along every point in the line P. This is again confirmed 
by the forms given to U and Y. For at any point where the coordinates are 
0, 0, ^, 9 the equations to the tangent planes to the two surfaces respectively 
are 

^x+6y = 0, 
c^a; + cOy = 0, 
that is to say, are identical. 

Whilst, therefore, certain grounds of geometrical, and still stronger 
grounds of analytical analogy, might seem to justify this species of contact 
taking the name of confluent quadrilinear, yet as, in fact, the intersection is 
trilinear, and as, moreover, the two indefinitely proximate lines must be con- 
sidered, not as coincident, but as turned away from one another through an 
indefinitely small angle and out of the same plane, I prefer to take advantage 
of this striking property of contact at every point along, a line (a property 
entirely distinct from any that we have yet considered), and confer upon the 
species of contact we have been considering the designation of unilinear- 
indefinite contact. 

Where the line of indefinite contact meets the two other lines of the 
intersection, the contact is of course of a Jiigher order ; thus offering a 
parallel to what takes place in ordinary unilinear contact, in which there 
is no contact, except only at two points of the right line forming part of the 
complete intersection. 

I believe that this kind of contact, which forms a natural family with two 
others about to be described, and which will close the list, has never before 
been imagined, and would at first sight have been rejected as impossible. 

Having now exhausted the cases of the first class, in which the minors 
have no factor in common, and the two sections of the second class, in which 
the second minors have no common factor, but the first minors of Z7+ XV a 
linear or quadratic function of \ in common, I descend to the third class, in 
which the second minors, which are quadratic functions of \, are supposed to 
have a common factor. 

This common factor must enter twice into each of the first minors by 
virtue of the law previously indicated, and cannot enter more than twice, as 



36] and Surfaces of the Second Order. 235 

otherwise the first minors of ?7 + 7^7" could only differ from one another by a 
numerical multiplier, which is obviously impossible, except when JJ -\-\y is 
of the form (h + A,) JJ, that is, when the two surfaces coincide. 

Again, the common factor of the first minor must enter three times into 
the complete determinant ; but there is no reason why it may not enter four 
times, and thus two cases arise. In the first, the characteristics take the 
form 

U =x-+if-^z--\- 1-, 

V = ax- + ay'- + az' + ht-. 
The second determinant having a factor in common, shows that the inter- 
section JJ, V is contained in a pair of coincident planes ; but the complete 
determinant, having two distinct factors, evidences that these plane inter- 
sections, viewed as indefinitely near but still distinct, lie in the same cone, 
which will be a cone enveloping both the surfaces JJ and V all along their 
mutual intersections. This is also seen easily from the forms of JJ and V; 
for we have V — aJJ = (b — a) t^, which proves that the intersection lies in 
the coincident, or, to speak more strictly, consecutive planes ff' = ; and at 
any point x = ^, y = r], z=^, the tangent plane to each surface becomes 

^x + vy + ^^ = 0- 

As there are six independent, that is, uon-necessarily co-evanescent second 
minors, that the second minor systems shall all have a common factor, implies 
the satisfaction of five conditions. This species of contact I call curvilineo- 
indefinite; it is, I believe, the only kind of indefinite contact between two 
surfaces of the second order hitherto taken account of. 

There is still, however, a higher species of contact, videlicet, when all the 
four roots of the complete determinant of JJ +XV are identical with the root 
common to each of its second minors. In this case the common enveloping 
cone becomes identical with the plane (considered as a coincident pair of 
planes) in which the surfaces intersect. 

The characteristics take the form 

JI = X- + xy -r zt, 

y = ^y + zt. 
The intersection is contained completely in the common tangent plane 
a; = 0, and consists of the two right lines, 

{x = 0, z = 0), 

{x = 0, t = 0). 

This, the highest and crowning species of contact, I call bilineo-indefinite. 
It is defined b}^ six conditions. 

At each point of the two lines of intersection of JI and V there is contact, 
and a very peculiar species of contact at the intersection of these two lines 
themselves. 



236 An Enumeration of the Contacts of Lines [36 

To form a distinct idea of this, let the physical visible or quasi-visible 
intersection of U, V take place along the two lines L, M ; the rational inter- 
section must be conceived as made up of the wry quadrilateral, L, M; L', M', 
in which L is indefinitely near to L', and M to M'. It follows, therefore, that 
there is contact at the four angles of the quadrilateral ; but as there is 
nothing to fix the relative directions of the diagonal joining the intersection 
of L and M to that of L' and M', because there is nothing to restrict the 
position of the latter point, except that it shall lie upon either surface*, it 
appears that not only is there contact at the junction of the two lines 
constituting the complete intersection of the two surfaces, but that these 
surfaces continue to touch at consecutive points taken all round this first, 
and indefinitely near to it in any direction f. 

Bilineo-iudefinite (the highest) contact for two conoids is strictly 
analogous to confluence, the highest species of contact between conies. 
For this latter may be conceived as an intersection made up of two co- 
incident pairs of coincident points ; and the former, as an intersection made 
up of two coincident pairs of crossing right lines ; and a pair of crossing 
lines is to a plane locus of the second degree what a coincident pair of points 
is to a rectilinear locus of the same degree. 

In the subjoined table I have brought under one point of view the 
characters and algebraic forms which I call the condensed forms corre- 
sponding to each species of contact above detailed. 

A. Quadratic loci in a right line. 
Simple contact. ] xy 

One condition. j x" + xy 

B. Quadratic loci in a plane. 
1st Class. 

Simple contact. ] x^ + y^ + ^^^ 

One condition. j ax'^ + 6?/ -I- cxz 

Proximal contact. 1 x^ + yx + yz 

Two conditions. j ax'' + byx + ayz 
2nd Class. 

Diploidal contact. | x" -\- y- + z"- 

Two conditions. J ax^ + ay- + hz'^ 

Confluent contact, j x"^ + y- + xz 

Three conditions. j y- + xz 

* This will be better seen by reference to the analogy presented by the case when the two 
conoids touch all along a curve. The rational intersection is made up of this curve and another 
indefinitely near it. The two curves, whatever be the position of their node, will lie in the same 
enveloping cone, so that the position of the node is indeterminate. 

f As the two surfaces jut one close into the other at this point, it would perhaps be not 
improper to designate the contact at such point as umbilical. 



36] 



and Surfaces of the Second Order. 



237 



C. Quadratic loci in space. 



1st Class. 

Simple contact. 
One condition. 

Proximal contact. 
Two conditions. 

Unilinear contact. 

1st species of diploidal. 

Two conditions. 

Confluent-unilinear, or 

triple contact. 
Three conditions. 

2nd Class, 1st Section. 

Biplanar contact. 

2nd species of diploidal. 

Two conditions. 



x^ + 'f -\- z"- + a:t 
ax- + by" + cz" + dxt 

x^ + y- + xt -V zt 
ax"' + by^ + cxt + azt 

x'^ + xy + zt \ 
ayz + bxy + czt i 



x" + yz + xy + zt \ 
az- + bxy + bzt ] 



x^ + y'' + z^ + f 
ax^ + ay"^ + hz- + ct" 



Confluent-biplanar con-) x' + zt + y'^+z- 



tact. Three conditions, j ax" + azt + by"- + cz 

x^ + y- + z^ + zt 
ax" + ay" 4- bz"^ + czt 



Bilinear contact. 
Three conditions. 

Confluent-bilinear con- 
tact. Four conditions. 

2nd Class, 2nd Section. 

Quadrilinear, or quad- 
ruple contact. 
Four conditions. 

Unilineo-indefinite con- 
tact. Five conditions. 

3rd Class. 

Curvilineo-indefinite 

contact. 
Five conditions. 

Bilineo-indefinite con- 
tact. Six conditions. 



xz + yt 
axt + byz 



xz + xt + y'^ -t- z"- 
axz + bxt + by" + bz^ 



x^ + y- + z^ + t^ 
ax^ + ay" + bz"- + bf 

x'' + y--^«:z + yt 
ax'' + by^ + cxz + cyt 



x^ + y"--{-z" + t" 
ax^ + ay" + az^ -t- bt- 



x'^ + ocy + zt 
xy + zt 



(xy + zt 
\ axy + bzt 



■238 An Enumeratio7i of the Contacts of Lines [36 

Another (and, in a physical sense, more) natural mode of grouping the 
twelve species of conoidal contact, which, without observing the same lines 
of demarcation, leaves intact the sequence of the species, is into the three 
families. The first, or definite-continuous, for which the surfaces touch in 
a single point, and intersect in an unbroken curve, comprises simple and 
'Cuspidal contact. 

The second definite-discontinuous, for which the surfaces touch in one, 
two, three or four points, but intersect in a curve more or less broken up into 
distinct parts, comprises all the species from the third to the ninth inclusive. 
The third natural family is that of indefinite contact, and comprises the three 
last species. It will of course be observed that there are five species of single 
contact, that is, contact at one point, namely, simple, cuspidal, and the three 
■confluent species, two of double, one of treble, one of quadruple, and three of 
indefinite contact ; the last being distinguishable inter se — lineo-indefinite as 
being special at two points, cuivilineo-indefinite as having no speciality, and 
bilineo-indefinite as being special at one point only. 

I might now proceed to discuss more particularly the nature of the 
■contact taken, not collectively, but with reference to each single point where 
it exists. This, however, must be reserved for a future communication ; as 
also, among other important and curious matter, the ascertainment of the 
singular forms of quadratic conjugate functions of five or more letters. At 
present I shall content myself with stating the following general proposition, 
which naturally suggests itself from a consideration of the cases already 
considered. 

In a conjugate quadratic system of any number of letters, the lowest and 
also the highest degree of singularity will be always unique ; the conditions 
to be satisfied in the former case being only one in number, and in the latter 
\r{r — 1), where r denotes the number of the letters. The first part of this 
proposition is self-apparent, the latter part may be inferred from the homa- 
loidal law ; for the (r — 2)nd minors will be quadratic functions, and the 
highest degree of contact will correspond to those having a factor in common, 
which would involve the satisfaction of \r{r — 1) — 1 conditions only; but 
over and above this, that the complete determinant, instead of containing 
this common factor, as it needs must, (r — 1) times, shall contain it r times : 
this gives one condition more, making up the entire number to ^r {i — 1). 

The total number of different species of singularity for conjugate func- 
tions of a given number of letters, can only be expressed by aid of formulae 
containing expressions for the number of various ways in which numbers 
admit of being broken up into a given number of parts. 

The computation of this number in particular cases, upon the principle of 
the foregoing method, is attended with no difficulty. 



36] and Surfaces of the Second Order. 239 

We have seen that this number for two, three and four letters, is 
respectively one, four, twelve. 

I have found that for five letters the number is twenty-four, for six 
letters fifty, for seven letters a hundred, and (subject to further examination) 
for eight letters one hundred and ninety-three. The series, therefore, as far 
as I have yet traced it, is 1, 4, 12, 24, 50, 100, 193. The last number must 
not be relied upon at present. 

It will be observed, that the foregoing table for the contacts of surfaces 
of the second order contains no form corresponding to a complete intersection 
in two non-intersecting lines and an undegenerated conic. In fact, if two 
such lines form part of the intersection, at least one other right line inter- 
secting them both, must go to make up the remaining part. This is easily 
verified ; for it is readily seen that the most general representation of two 
conoids intersecting in two non-meeting lines will be 

U= xy + zt, 

F= axy + hzt + cxt + eyz, 

where the two lines in question are 

{x = 0, z = 0), 

(2/=0, i=0). 

Now it will be found that the first minors of V+XU formed from the 
above equation will all contain the common factor {a + \) (b + X) — ce, showing 
that the contact is quadrilinear or linear-indefinite, that is bilinear, according 
as the roots of 

X^ + {a + b)\+ab-ce = 

are distinct or equal ; which explains how it is that only one species of 
bilinear contact (that is to say, the case corresponding to the two conoids 
agreeing in the two right' lines in which each is cut by a common tangent 
plane) comes to find a place in the preceding enumeration. 

It may not be uninteresting, under an euristic point of view, to state that 
the above theory, which, as well in what it accomplishes as in what it 
suggests (the author cannot but feel conscious), constitutes a substantial 
accession to analytical science, arose out of a theorem which occurred to 
him as likely to be true, in the act of reviewing for the press his paper 
"On Certain Additions " in the last November Number* of this Magazine, 
and which he had only then time to throw into a foot-note as a probable 
conjecture. 

Wishing to subject it to an analytical test, he found it necessary to obtain 
the condensed forms which serve to characterize the confluent contact of 
[* p. 148 above.] 



240 Contacts of Lines and Surfaces of the Second Order. [36 

conies. In this way he became aware of the great utility of these condensed 
forms, and of the desideratum to be supplied in obtaining a complete list of 
them applicable to all varieties of contact. The happy thought then occurred 
to him of inverting the process which he had applied in the treatment of 
the contacts of conies, in the November Number* of the Gavibridge and 
Dublin Mathematical Journal ; for whereas the nature of the contacts was 
there assumed and translated into the language of determinants, he soon 
discovered that it was the more easy and secure course to assume the relations 
of every possible immutable kind that could exist between the complete and 
minor determinants corresponding to the characteristics, by aid of these 
relations to construct the characteristics, and from the characteristics so 
obtained, determine the geometrical character of each resulting species of 
contact. Thus he has been able to effect the very results stated by himself 
as desiderata at the close of the paper in this Magazine above referred to. 

Note. — It is proper to remark, that all the condensed forms given in this 
paper have actually been obtained by the author in the way above pointed 
out. The limits imposed by the objects to which the Magazine is devoted 
have restricted him from exhibiting the method at full ; but any of his 
readers will be able without difficulty to make it out for himself. 

The process consists in finding JJ+Why means of solving for each case 
a problem of position (a kind of chess-board problem) on a square table, 
containing three places in length and breadth for conies, four places by four 
for surfaces, and so on (if need be) according to the number of variable letters 
involved. CT+XF being thus determined in form, ?7 and F become readily 
cognizable. It is right also to add, that some of the condensed forms here 
set forth have been incidentally noticed and employed by previous authors, 
as Pliicker and Mr Cayley. 

The conditions in each case to which the position-problem is subject 
are immediately deducible from the laws which the complete determinant, 
and the successive minor systems of determinants of U+XV, are required 
to satisfy. 

[* p. 119 above.] 



s^ 



37. 



ON THE RELATION BETWEEN THE MINOR DETERMINANTS 
OF LINEARLY EQUIVALENT QUADRATIC FUNCTIONS. 

[Philosophical Magazine, I. (1851), pp. 295—305.] 

I SHOWED in the preliminary part of my paper on Contacts in the February 
Number of this Magazine*, by a priori reasoning, that if a quadratic function 
{V) be linearly converted into another {V), any minor determinant of any 
order of Fmust be a syzygetic function of all the minor determinants of U of 
the same order. 

The object of my present communication is to exhibit the syzygy in 
question, which, as I indicated, is linear ; by which I mean that a determinant 
of the one function is equal to the sum of the pari-ordinal determinants 
of the other affected respectively with multipliers formed exclusively out of 
the coefficients of the equations of transformation. In order that a clear 
enunciation of the theorem in view may be possible, it is necessary to premise 
a new but simple, and, as experience has proved to me, a most powerful, 
because natural, method of notation applicable to all questions concerning 
determinants. 

Every determinant is obtained by operating upon a square array of 
quantities, which, according to the ordinary method, might be denoted 
ias follows : 



■ ■ai,i 



' My method consists in expressing the same quantities biliterally as 

illow : 

V aia^, a-i^ct^ ... OiCin, 

I aaOti, a2«2 ... a^an, 



«««!, a^a^ . . . anO-n, 
[* p. 221 above.] 



16 



242 The Relation between the Minor Determinants of [37 

where of course, whenever desirable, instead of a,, fla--- ««> and a.-^, a„ ... «„, 
we may write simply a, b ... I, and a, ^ ... X respectively. Each quantity is 
now represented by two letters; the letters themselves, taken separately, 
being symbols neither of quantity nor of operation, but mere umbrae or ideal 
elements of quantitative symbols. We have now a means of representing 
the determinant above given in a compact form ; for this purpose we need 

but to write one set of umbrae over the other as follows: ( ^' " '" ") . If 



we now wish to obtain the algebraic value of this determinant, it is only 
necessary to take a^, a^ ... o'ni'o. all its 1, 2, 3 ... n different positions, and we 
shall have 



fa, , a^ ... an] ^ , 

\ ^ = i + \a,ag^ X a^oig. x ... x a,. 



in which expression 6^, d^.-.dn represents some order of the numbers 
1, 2 ... n, and the positive or negative sign is to be taken according to the 
well-known dichotomous law. Thus, for example, 

■^ >• will represent aa x b/3 x cy ' 
+ a/3 X 67 X ca 
+ a^ X 6a X c/3 

— a^xhoL X cy 

— aa xby x c/3 

— ay X b/3 X ca 

Although not necessary for our immediate object, it may not be inop- 
portune to observe how readily this notation lends itself to a further natural 
extension of its application. 

a& cd\ will naturally denote 
a/S yh] 

ab cd ab cd 

a^ yS 78 a/3' 

{aoLxbl3)\ j {cyxdS)l_( (ayxbS)] ( (caxd^) 
■ (a/3 X ba)\ ^ \- (cS x dy)] \- (aS x by)] "^ [- (c/3 x da) 

And in general the compound determinant 



that is 



will denote 

2± 



li, 61 ... ^1, aa, 62 ••• 4 ••• «r. br ... lr\ 
?l, /3i...\i, tta, /S2...X2 a-r, /3r...Xr) 



a,, b, ... L) (a,, bo ... L] far, l>i- ••• ^r 

r X -l ~ r X X •{ 



37] Linearly Equivalent Quadratic Functions. 243 

where, as before, we have the disjunctive equation 

e„ e._...er = i, 2...r. 

As an example of the power of this notation, I will content myself with 
stating the following remarkable theorem in compound determinants, one of 
the most prolific in results of any with which I am acquainted, but which 
is derived from a more particular case of another vastly more general. The 
theorem is contained in the annexed equation 



..Ur, Ur+i, Oil, CC«...(Xr, ttr+a «!, Ct^.-.d^, Q 

«!, O2 ... ay]*'"^^ ftti, 0.2... ar, Ctr+i, ar+2---«r+s 
0.1, dz ... Clr, «r+l J *r+2 • • ■ ^^r+s. 



(1) 



It is obvious, that, without the aid of my system of umbral or biliteral 
notation, this important theorem could not be made the subject of statement 
without an enormous periphrasis, and could never have been made the object 
of distinct contemplation or proof. 

To return to the more immediate object of this communication, suppose 
that we have any binary function of two sets of quantities, x^, x^-.-cCn] 
^1, ^2 •■■^11, of which the general term will be of the form Cr^s")^ Xr^s', 
according to the principles of notation above laid down, nothing can be 
more natural than to represent c^.s by the biliteral group 0^0^; the function 
in question will then take the form 

the x's and |^'s denoting quantities, but the a's and a's mere umbriE. The 
function may then be thrown under the convenient symbolical form 

(OiaJi + tts^'a + • • • + ^n ^n) 

I So if we confine ourselves to quadratic functions, for which x-^, x^...Xn; 
^i,^2---^n become respectively identical, the general symbolical represen- 
tation of any such will be 

(a^Xi + a^x^ + . . . 4- a^Xny. 

The complete determinant will be denoted by 

tti, eta ••• (^n\ 
«!, Ha ... On) 

and any minor determinant of the rth order by 



16—2 



244 The Relation between the Minor Determinants of [37 

where 6^, 6^ ... 6r are some certain r distinct numbers taken out of the series 
1, 2, 3 ... r. Suppose now that we have 

U = (a^Xi + CI2OC2 + . . . + anXnY 

linearly transformable into 

V = {hy, + b,y,+ ...+bnyn)% 

by means of the n equations 

«i = ai6i . 2/1 + fii^s . 2/2 + . . . + a^bn . y, 
Xo = anby . y-i + a^bo . 2/2 + • . • + a^bn . y,. 



Xn = a-iA ■ yx + ««^2 . 2/2 + • • • + a«&« • yn 

in which equations, be it observed, each coefficient a,.6s is a single quantity, 
perfectly independent of the quantities denoted generally by a^a^, brbg which 
enter into U and V. Our object is to be able to express the minor 
determinant 

\bi,, bi^...bi, 

in which the one group of distinct numbers, ki,k^...k,. may either differ 
wholly from, or agree wholly or in part with the other group of distinct 
numbers l^, l^ ...Ir, under the form of 

2; (/'««.. «fl2--- «flr 

(Off ff \ 

rh' 1 " (h)' 
,a a a \ 

may be denoted by Q ( ," ^ '" ,'J ; so that our problem consists in deter- 

,a a a \ 

mining the value of Q l]^' ?'" ^j in the equation 

h,, bi^... bij \ \4>i' (j}2--(l>J \a^x, a*2 

Accordingly I enunciate that 

„ /^,, 0^ ... dr\^ faj,.,, a^^ ... ffi J ^ fa;,, a,, ... a,; 
^ [<f>„ cj,,... (pj K, be, ... bej K„ &*3 ... b. 

[be^, be,...be^\ [b^^, b^^ ... VJ 

subject to one sole exception in the case of 6,, 6„ ... 0r being identical with 
01, cf),, ... <f)r', namely, that for the terms (for such case) of the form 



37] Linearly Equivalent Quadratic Functions. 245 

Q\ a' a'a)' ^^^ value to be taken is not that which the general formula 
would give, namely, 

2 K' O'k.-.-akr- 
\he,, hg^... 6eJ ' 

but the half of this, that is simply the square of 
at,, fli, ... ai, 
ei , flea ■ • ■ ^fl, 

The value of Q ( ." ,'"".'), it is obvious, contains only quantities 

of the form ar.hs, which are coefficients in the equations of transformation, 
but none of the form a^ . a^ or br-bg; showing that the syzygetic connexion 
between the minor determinants of U and V of the same order is linear, 
as has been already anticipatively announced. 

The problem which I have treated above is only a particular case of 
a more general one, which may be stated as follows : given 

U= (a^oci + a.20i!.2 + . . . + anXny, 

and supposing m linear equations to be instituted between x^, x„...Xn, 
so that U may be made a function of {n — m) letters only, to express any 
minor determinant of the reduced form of U without performing the process 
of elimination between the given equations. Let the given equations be 
written under the form 

aia„+i3?i + aMn+iX^ + ... + a,ia»i+i«n = 0, 
a-fln+^x-^ + a^an^-iXn + ... 4- a,ia„+2ir„ = 0, 



, nd let it be convened (which takes nothing away from the generality of 
t lese equations) that an+rO-n+s shall signify zero for all values of r and s 
c ncurrently greater than zero. Suppose that x^, x.,...Xm, being eliminated, 
L becomes of the form 

i' \Pm+i^m+i "I Om+2^m+2 + ■ ■ ■ + OnXn) j 

and suppose that we wish to determine the value of the complete determinant 
of this last function ; it will be found to be 

J.1, bm+2 ••• M _ j«i. a-i--- dn, an+i ... Ctn+ml ^ f<h., ^2 ••• ^n 
Pm+i> bjn+2 ■■•(>„/ [(hi 0,2...Cln> K»+i • • • O^n-l-mj V^n-i-i, (Xn+2 . . . CSji+m 

the squared divisor being, as is obvious, a function only of the coefficients 
of the transforming equations, and depending for its value upon the particular 



246 The Relation between the Minor Determinants of [37 

m quantities selected for elimination. The dividend, on the contrary, is 
independent of this selection, but involves the coefficients of the function 
combined with the coefficients of transformation. This is the- sj'mbolical 
representation of the theorem given by me in the postscript to my paper in 
the Ca7nbridge and Dublin Mathematical Journal for November 1850*. 

Suppose, now, more generally that we wish to find any minor determinant. 
The solution is given f by the equation 

l^^m+l' hm+2 ■■• hm+s) 

(wherein the two groups dm+i, 6m+2, ■•■^m+s', <l>m+i, 'f>m+2 ■ ■ ■ <jim+s are each 
of them s differing, or wholly or in part agreeing individuals arbitrarily 
selected out of the {n — in) numbers m + 1, m + 2, ... n) 

If we make n = 2^ and m = y, and ay+rdy+s — for all positive values 
of either r or s, and ay_ia,j+e= for all values of i and e differing from one 
another, and for equal values ay^e^h+c — ~ 1' i* ^'"^ readily be seen that this 
last theorem reduces to the one first considered ; and on careful inspection 
it will be found, that the solution given of the general question includes 
within it that presented for the particular case in question. Such inclusion, 
however, I ought in fairness to state is far from being obvious; and to 
demonstrate it exactly, and in general terms, requires the aid of methods 
which my readers would probably find to exceed their existing degree of 
knowledge or familiarity with the subject. 

The theorem above enunciated was in part suggested in the course of a 
conversation with Mr Cayley (to whom I am indebted for my restoration 
to the enjoyment of mathematical life) on the subject of one of the pre-/ 
liminary theorems in my paper on Contacts in this Magazine. 

It is wonderful that a theory so purely analytical should originate i: 
a geometrical specula.tion. My friend M. Herniite has pointed out to mf,', 
that some faint indications of the same theory may be found in the Recherchi j 
Arithmetiques of Gauss. The notation which I have employed for dete, - 
minants is very similar to that of Vandermonde, with which I have becori, e 
acquainted since writing the above, in Mr Spottiswoode's valuable treatis'3 
On the Elementary Theorems of Determinants. Vandermonde was evidently 
on the right road. I do not hesitate to affirm, that the superiority of his 
and my notation over that in use in the ordinary methods is as great and 
almost as important to the progress of analysis, as the superiority of the 
notation of the differential calculus over that of the fiuxional system. For 
what is the theory of determinants ? It is an algebra upon algebra ; a 

[* p. 136 above.] [t see p. 251 below.] 



37] Linearly Equivalent Quadratic Functions. 247 

calculus which enables us to combine and foretell the results of algebraical 
operations, in the same way as algebra itself enables us to dispense with 
the performance of the special operations of arithmetic. All analysis must 
ultimately clothe itself under this form*. 

I have in previous papers defined a " Matrix " as a rectangular array of 
terms, out of which different systems of determinants may be engendered, 
as from the womb of a common parent ; these cognate determinants being 
by no means isolated in their relations to one another, but subject to certain 
simple laws of mutual dependence and simultaneous deperition. The con- 
densed representation of any such Matrix, according to my improved Vander- 
mondian notation, will be 

a2 ■ • • "mj ' 

To return to the theorems nf the text. Theorem (2) admits of being 
presented in a more convenient form for the purposes of analytical operation, 
so as to become relieved from all cases of exception appertaining to particular 
terms. 

The limitation to the generality of the expression for Q arises from our 
treating 

\ as identical with its equal, 

I If, however, we now convene to treat these two forms as distinct, so that 
in theorem (2) 

\ (n(n-l)...{n-r+l)\\ ^, -^ • i 

will contain ^^ — ~ '\ terms, then we may write simply 

^ „ /^i, d^ ... 0r\ ^ (aij, ttij ... aj ^ <^ai^, a^ ... ai\ 

I * Perhaps the most remarkable indirect question to which the method of determinants has 
ibeen hitherto applied is Hesse's problem of reducing a cubic function of 3 letters to another 
consisting only of 4 terms by linear substitutions — a problem which appears to set at defiance 
all the processes and artifices of common algebra. I have succeeded in applying a method 
founded upon this calculus to the linear reduction of a biquadratic function of two letters to 
Cayley's form x' + mx-y^ + y*, and of a 5' function of two letters to the new form x^ + y^ + (ax + by)^. 
This last reduction is effected by means of the properties of a certain other function of the 
8th degree connected with the given function of the 5th degree. See a paper on this subject in 
the forthcoming May Number of the Cambridge and Dublin Mathematical Journal, [p. 191 above.] 



248 The Relation hetween the Minor Determinants of [37 

which equation is subject to no exception for the case of the ^'s and ^'s 
becoming identical. As regards this theorem, it will not fail to strike the 
reader that it ought to admit of verification ; for that TJ may be derived 
from Fin the same manner as F from JJ if we express yi, y^...yn in terms 
of Xi, x^...Xn, by solving the system of equations (2), which there is no 
difficulty in doing. In fact, if we write 

yi = (lS^x^ + ai/32 a;., + . . . + a-,^n ^n, 

IJo = aa/Si^i + Oo/Sa X„_-V ...-\- OiSn ^n . 



we shall obtain 



arP.= 



yn = «n/3iO0i + a„/3o X2 + ...+ Un/Sn Xn, 



bi, bo... bs-1, &S+1. is+2 •■• i>n] ' \bi, bo ... bj ' 



Accordingly we shall find 

\ap^, aj,^...apj \ Wi, ao_...a,J {b^^, b^^ ... b^, 
and 

/i/ti, fo... ^lr,.\ ^ fa,n^, a™, ... «™,.\ ^ /«,,, a^i, ••• «i) 
^Ui, coo_...corJ V/3^,, I3^^.../3^J V/3„i, /S„,.../S<„ 

substituting for the a's and yQ's their symbolical equivalents given above, 
and applying the theorem given below, we shall easily obtain 

(Ul, (0.2 ... CoJ \b^,.+i, b^^^^...b^,J \^pr+i, ^p,+2--- ^Pn 

ja.i, fla ••• «)i]" 
[6,, &2 ... bn] ■ 

If, now, in the expression 

hi, h^--- K\ ^ v [C^h' "■h--- «*r\ f^h, ah--- ai\ fa^^, Ue, ... ag\ 
hi , h^...bi^\ "* lUe, , ie^ . . ■ be J U^, , b^.^ . . . b^J U^, , a^^... a^J 

we resubstitute for -^ *'' ^ '"' *"■[ its value in the form of 

^{(b^i,h,-hr' 

we shall obtain ( , ' ' , '^ ' " t*' ) under the form of 
\b^, bi^... bJ 

\.-«|fi, ■^^...■^r' \hi' '^i2 



37] Linearly Equivalent Quadratic Functions. 249 

and rC^,^' "/'■■■ /) must =0, except for the case of eoj, toj.-.wr; 1^1, •<i/-2---'>f'r 

being respectively identical with ki, k^.-.K; li, l^-.-lr, for which case 

fk k k \ 
jR l^' ''"' '') must be unity. I have gone through this calculation and 

\6j , ^2 • • • 'r/ 

verified the result ; in order to effect which, however, the following important 
generalization of theorem (1) must be apprehended. 

Suppose two sets of umbrae, 

(Xj , CI2 . . . O^m+n ) 
Oi , O2 • • ■ Om+n ! 

and let r be any number less than m, and let any ?'-ary combination of 
the m numbers 1, 2, 3 ...m be expressed by «^i, 10^... 10^, where q goes 
through all the values intermediate between 1 and fi, fi being 

m (m — 1) . . . (m — r + 1) 
1.2 ...r ' 

then I say that the compound determinant. 



"^'^i' '^iSj '" ^^im' '^™+i' f'ni+2 ••• ^m+n| ^2^ > C^aj ••• ^^Bm' '^'n+i' '^'n+2 ••• '^m+il 



^^0 ) ^>ie^ * • ■ ^/'Sni ' ^m+i ) ^^?n+2 ' • ■ ^^m+n | 

is equal to the following product, 



\ One, > '^1^02 • ■ ■ I^Sm ' "»+i ' ''"i+s • • • Om+n > 



irhere 



,1 _ (m— l)(?w — 2) ... (m — r + 1) 
^ ~ 1.2...(r-l) ' 

, {m — 1 ) (m — 2) . . . («i — r) 



(4) 



H- 



1.2...r 



■When r = 1, we have the case already given in theorem (2), and of course 
/i" is to be taken unity. 

This very general theorem is itself several degrees removed from my still 
unpublished Fundamental Theorem which is a theorem for the expansion 
of the products of determinants. 



250 On Linearly Eqtiivalent Quadratic Functions. [37 

Obs. The analogy upon which the extension of the Vandermondian 
notation from sitnple to compound determinants is grounded, would be better 
apprehended if the biliteral symbols of simple quantities were written with 

the umbral elements disposed vertically, as , , instead of horizontally, as ah ; 

which latter is the method for the purposes of typographical uniformity 
adopted in the text above. The other mode is, however, much to be pre- 
ferred, and is what I propose hereafter to adhere to. For my two general 
umbrae, a, b, Vandermonde uses two numbers, one set a-cock upon the other, 
as 5*. The objection to the use of numbers is apparent as soon as it becomes 
necessary to treat of the mutual relations of diverse systems of determinants, 
and his mode of writing the umbrae militates against the perception of the 
most valuable algebraical analogies. The one important point in which 
Vandermonde has anticipated me, consists in expressing a simple determinant 
by two horizontal rows of umbrfe one over the other. But the idea upon 
which this depends is so simple and natural, that it was sure to reappear 
in any well-constructed system of notation. 



38. 



NOTE ON QUADRATIC FUNCTIONS AND HYPER- 
DETERMINANTS. 

[Philosophical Magazine, i. (1851), p. 415.] 

Permit me to correct an error of transcription in the MS. of my paper 
" On Linearly Equivalent Quadratic Functions " in the last number of the 
Magazine. The theorem [p. 246 above] marked (3), should read as follows : — 

>! {"e„M' "e»i+2 • • • ^em+« 

tti, Clj • • • ^7n ! OSflm+i > ^em+2 ' ' ' '^^m+s > ^n+l I (^11+2 • ■ • 0,n+m I 

«! , (I2 . . . am , 0!.|j>m+i J '^4'm+2 ■ • ■ '■Hm+s > *»+i ' '*«+2 ■ ■ • Ctm+m J 
«!, tta ... dm 1^ 

I I may take this opportunity of mentioning, that by extending to 

algebraical functions generally a multiliteral system of umbral notation, 

'1 analogous to the biliteral system explained in the paper above referred 

Ii to as applicable to quadratic functions, I have succeeded in reducing to 

j , a mechanical method of compound permutation the process for the discovery 

ll of those memorable forms invented by Mr Cayley, and named by him hyper- 

\ determinants, which have attracted the notice and just admiration of analysts 

all over Europe, and which will remain a perpetual memorial, as long as 

[ the name of algebra survives, of the penetration and sagacity of their author. 



39. 



ON A CERTAIN FUNDAMENTAL THEOREM OF 
DETERMINANTS. 

[Philosophical Magazine, II. (1851), pp. 142 — 14.5.] 

The subjoined theoi'em, which is one susceptible of great extension and 
generalization, appears to me, and indeed from use and acquaintance (it 
having been long in my possession) I know to be so important and funda- 
mental, as to induce me to extract it from a mass of memoranda on the same 
subject; and as an act of duty to my fellow-labourers in the theory of 
determinants, more or less forestall time (the sure discoverer of truth) by 
placing it without further delay on record in the pages of this Magazine. Its 
developments and applications must be reserved for a more convenient 
occasion, when the interest in the New Algebra (for such, truly, it is the 
office of the theory of determinants to establish), and the number of its 
disciples in this country, shall have received their destined augmentation. In 
a recent letter to me, M. Hermite well alludes to the theory of determinants 
as " That vast theory, transcendental in point of difficulty, elementary in 
regard to its being the basis of researches in the higher arithmetic and in 
analytical geometry." 

The theorem is as follows : — Suppose that there are two determinants of 
the ordinary kind, each expressed by a square array of terms made up 
of n lines and n columns, so that in each square there are ?i^ terras. Now 
let n be broken up in any given manner into two parts p and q, so that 
p + q = n. Let, firstly, one of the two given squares be divided in a given 
definite manner into two parts, one containing p of the n given lines, and the 
other part q of the same; and secondly, let the other of the two given squares 
be divided in every possible way into two parts, consisting of q and p lines 
respectively, so that on tacking on the part containing q lines of the second 
square to the part containing p lines of the first square, and the part con- 
taining p lines of the second square to the part containing q of the first, we 



39] On a Fundamental Theorem of Determinants. 253 

get back a new couple of squares, each denoting a determinant different 
from the two given determinants ; the number of such new couples will 
evidently be 

n{n— 1) . . . (?i — p + 1) 
1.2... p ' 

and my theorem is, that the product of the given couple of determinants 
is equal to the sum of the products {affected with the propter algebraical sign) 
of each of the new couples formed as above described. Analj'tically the theorem 
may be stated as follows. 



according to the notation heretofore* employed by me in the preceding 
numbers of this Magazine, denote any two common determinants, each of 
the nth order, and let the numbers 6^, 6^ ■■■ ^n be disjunctively equal to the 
numbers 1, 2 ... n and jo + g' = ?i; then will 



\h, 6,...&J^lA, /8,...A. 



[ii, h...bp, ^gj,^^, ^g^^^...^ej i/3e,, ^«.,.../3ej„ bp+„ bp+^... 

The general term under the sign of summation may be represented by aid 
of the disjunctive equations 

(^1, (p2... (f>n=l, 2 ... 71, 
i/fi, i/r2...i|r„ = l, 2 ... n, 

under the form of 

(a^j .b^xa^„.b„x ... x a^^ . b^) (a^p^, . bp+^ x a^^^^ .bj,^^x ... xa^^. bn) 

X («*p+i • /Se^+i >^ «*p+2 ■ /3e^2 x . . . x a^„ . /S^J (=r^, . 3e, x a^^.^^^x ... x a^^ . A^,). 

1st. When ^i, ^2--- 4>p = '^i> i''2---'^p< it will readily be seen, that for 
given values of (^, , <^„...<^p, the product of the third and fourth factors 
becomes substantially identical with the general term of the determinant 

, Ota • • • «ii' 

and consequently, making the system (^i, ip^--- 4'p ('^^> '^^hich is the same 
thing, its equivalent yjri, i|/-2 ... ■yjrp) go through all its values, we get back for 
the sum of the terms corresponding to the equation 

^1, ^2 ... 4>p = -^i, ^2 ■■■ i^p, 

[* p. 242 above.] , 



254 On a certain Fundamental f39 

the product of the determinants 

^^' J-«4and^';- 

Ol, 60... hn) [Pi, /32... 

2ud. When we have not the equality above supposed between the <^'s 

and the -v^'s, let 

4>p-h = "^p+k and <^p+, = i|r^_f ; 

the corresponding term included under the S will contain the factor 

Now leaving </>!, (^o...<^p, and -v/fi, -^.^...-^^^ unaltered, we may. take a 
system of values 6-^, 6.! . . . &„', such that 

and ^ p-i = "p+vi ■ 

and for all other values of q except p+v, or p - ?, d'q = Oq- The correspond- 
ing new value of the general term so formed by the substitution of the 
6' for the 6 series, will be identical with that of the term first spoken of, but 
will have the contrary algebraical sign, because the 6' arrangement of the 
figures 1, 2, 3 ... p is deducible by a single interchange from the 6 arrange- 
ment of the same, the rule for the imposition of the algebraical sign plus 
or minus being understood to be, that the term in which 

/3ep+i, ^«;,+2 ■ • • /3e„ ; Pei > Pe^--- Pep 

enter into the symbolical forms of the respective derived couples of deter- 
minants, has the same sign as, or the contrary sign to, that in which 

/Saw /3eW2---^9'n; I3sv ^e',--/3e'p 
so enter, according as an odd or an even number of interchanges is required 
to transform the arrangement 



into the arrangement 

S'p+i, 0'p+2---0'n; O'l, 0\...d'p. 

I have therefore shown that all the terms arising from the expansion of 
the products included under the sign of summation, for which the disjunctive 
identity ^], A^... </)p = '^i, ■vlf 2 ■ • • '>/^p does not exist, enter into the final sum in 
pairs, equal in quantity and differing in sign, which consequently mutually 
destroy, and that the terms for which the said identity does exist together 
make up the sum 

j'tti, Oj ... «„! ( «!, ao ... 



39] Theorem of Determinants. 255 

which proves, upon first principles drawn direct from that notion of polar 
dichotomy of permutation systems which rests at the bottom of the whole 
theory of the subject, the fundamental, and, as I believe, perfectly new 
theorem, which it is the object of this communication to establish. 

In applying the theorem thus analytically formulized, it is of course to be 
understood that, under the sign 2, permutations within the separate parts of 
a given arrangement, 

are inadmissible, the total number of terms so included being restricted to 

m(w — 1) ... (n—p+ 1) 
1. 2...p ■ 

The theorem may be extended so as to become a theorem for the ex- 
pansion of the product of any number of determinants, and adapted so as to 
take in that far more general class of functions known to Mr Cayley and 
myself under the new name of commutants, of which determinants present 
only a particular, and that the most limited instance. 



40. 



ON EXTENSIONS OF THE DIALYTIC METHOD OF 
ELIMINATION. 

[Philosophical Magazine, ii. (1851), pp. 221 — 230.] 

The theory about to be described is a natural extension of the method of 
elimination presented by me ten years ago (in June, 1841) in the pages of 
this Magazine, which I have been induced to review in consequence of the 
flattering interest recently expressed in the subject by my friend M. Terquem, 
and some other continental mathematicians, and because of the importance 
of the geometrical and other applications of which it admits, and of the 
inquiries to which it indirectly gives rise. We shall be concerned in the 
following discussion with systems of homogeneous rational integral functions 
of a peculiar form, to which for present purposes I propose to give the name 
of aggregative functions, consisting of ordinary homogeneous functions of the 
same variables but of different degrees, brought together into one sum made 
homogeneous by means of powers of new variables entering factorially. 

Thus if F, 0, H ... L he any number of functions of any number of letters 
«, y ... i of the degrees ni, in — i, m — i.' . . . m — (t) respectively, 

i^+(?V + ir/ + ...+Z(9'" 

will be an aggregative function of the variables entering into F, G, &c. and of 
X, fi ... 6. I shall further call such a function binary, ternary, quaternary I 
and so forth, according to the number of variables contained in the functions ', 
F, G, H, &c. thus brought into coalition. 



It will be convenient to recall the attention of the reader to the meaninga, 
of some of the terms employed by me in the paper above referred to. / 

If F be any homogeneous function of x,y,z...t, the term augmentative i 
of F denotes any function obtained from F of the form | 

x'^y^zy ...fxF. ^ 

Again, if we have any number of such functions F, G, H ... K oi as many 



40] Extensions of the Dialytic Method of Elimination. 257 



variables x,y,z...t, and we decompose F, G, H ...Km any manner so as to 
obtain the equations 

F = ««P, + y''F, + z^F, + &c. . . . + t<i{F), 

G=x''Q, + y'>Q, + z^Q, + &c. ...+t^(Q), 

H = x^R, + y''R, + s'R, + &c. . . . + f^iR), 



K = a^S^ + y''S, + z'S, + &c. ...+^^(8), 

and then form the determinant 

A, F,, F,...(F) 

Qi, Q., Qs ... (Q) 
Ri, i?2, -Rs ... (R) 



Si, S^, Ss ... (S) 

this determinant, expressed as a function of x, y, z ... t, is what, in the paper 
referred to, I called a secondary derivee, but which for the future I shall cite 
by the more concise and expressive name of a connective of the system of 
functions F, G, H ... K from which it is obtained. One prevailing principle 
regulates all the cases treated of in this and the antecedent memoir, namely 
that of forming linearly independent systems of augmentatives or connectives, 
or both, of the given system whose resultant is to be found, of the same 
degree one with the other, and equal in number (when this admits of being 
done) to the number of distinct terms in the functions thus formed. The 
resultant of these functions, treated as linear functions of the several 
combinations of powers of the variables in each term, will then be the 
resultant of the given system clear of all irrelevant factors. If the number 
of terms to be eliminated exceed the number of the functions, the elimination 
of course cannot be executed. If the contrary be the case, but the equality 
is restored by the rejection of a certain number of the equations, the resultant 
so obtained will vary according to the choice of the equations retained for 
the purpose of the elimination. The true resultant will not then coincide 
with any of the resultants so obtained, but will enter as a common factor into 
them all. 

The following simple arithmetical principles will be found applicable and 
useful for quotation in the sequel : — 

(a) The number of terms in a homogeneous function of p letters of the 
mth degree is 

m (m + 1) . . . (m + p — 1) 
1.2...P ■ 

s. 17 



258 On Extensions of [40 

(h) The number of augmentatives of the {m + n)th degree belonging to 
a function of p letters of the mth degree is 

(?i + l)(w + 2) ...{n+p- 1) 
1.2. ..p 

(c) The number of solutions in integers (excluding zeros) of the equation. 
«! + do + ... +ap = k is 

{k-l){k-2)...{k-p + l) 
1.2...(^-1) 

To begin with the case of binary aggregatives. Let 

-Fm {OS, y) + -fm-. («, y) X' + F,^,. {x, y) ya'' + &c. . . . + F^^,) {x, y) 6"" 

Gn («, y) + <?«-. {a>, y) X' + Gn-,' {X, y) /X'' + &C. . . . + G'n-io («, y) ^l"' I ,^. 

Z'^ (a;, 2/) + Kp_, {x, y) A,' + Kp_,. (x, y) fi'' + &c. . . , + Zj,_,,) {x, y) 6 '" 

be a system of functions (whose Kesultant it is proposed to determine) equal 
in number to the variables x, y,X, fj. ... 9, and similarly aggregative, that is 
having only the same powers of X., fi, &c. entering into them, but of any 
degrees equal or unequal m,n ... p. Let the number of the functions be r. 
Raise each of the given functions by augmentation to the degree s, where 

s = (m + ?i + ... +p)-(t+ t' + ...+(t))-l, 

the number of augmentatives of the several functions will be 

(s + 1) — in, 

(s+ l)-n, 



{s+l)-p, 

aud the total number will therefore be 

r (s + 1) — (m + n+ ...-^-p), 

which =(r — l)(m + ?i+ ... +^) — r (i + t' + ... + {i)). 

Again, the number of terms to be eliminated will be the sum of the 
numbers of terms in functions respectively of the sth, (s — t)th, (s — t')th, . . . 
(s — (t))th degrees, which are respectively 

s + 1, 

s + 1 - 1, 

s + 1 — t', 

s + l- (0, 



40] the Dialytic Method of Elimination. 259 

and the number of these partial functions is r— 1. Hence the number of 
terms to be eliminated is 

(r - 1) {m + m + &c. + p - (t + t' + &c. + (t))} - (t + t' + &c. + (0) 

= (r- l)(?w, + M + &c. +^)-r(t + t'+ ... +(0), 

which is exactly equal to the number of the augmentative functions. Hence 
the Resultant* of the given functions can be found dialytically by linear 
elimination, and the exponent of its dimensions in respect to the coefBcients 
of the given functions will be the number 

(r — 1) 2m — rSt, 
as above found. 

The method above given may be replaced by another more compendious, 
and analogous to that known by the name of Bezout's abridged method for 
ordinary functions of two letters. As the method is precisely the same 
whatever the number of the functions employed may be, I shall for the sake 
of greater simplicity restrict the demonstration to the case of three functions, 
U, V, W, whose degrees (if unequal, written in ascending order of magnitude) 
are m, n, p respectively. Let 

U = F^{x, y) + Fr,^, {x, y) z\ 

F= Gn («, y) + G'^. {so, y) z', 

W = Rp (x, y) + Hp_, (x, y) zK 

Let 6, Q) be taken any two numbers which satisfy in integers greater than 
zero the equation + ca = m + 1, and let 

F„ (x, y) = (j}^g. x^ + (^^„ . y, 

Gn («. y) = 7«-e • ^^ + in-^ ■ y", 

Sp {oo, y) = Tip-e . of + r;j,_„ . y, 
where the <^'s, 7's, 7;'s may be always considered rational integer functions of 
X and y\ for every term in each of the functions F^m, Gn, Hp must either 
contain x^ or j/", since, if not, its dimensions in x and y would not exceed 

that is m — 1, whereas each term is of m conjoined dimensions, at least, in x 
and y. Hence from the equations 

U = Q, 

F = 0, 

F=0, 

* The Eesultant of a system of functions means in general the same thing as the left-hand 
side of the final equation (clear of extraneous factors) resulting from the elimination of the 
variables between the equations formed by equating the said functions severally to zeiw. 

17—2 



260 On Extensions- of [40 

by eliminatiug of, y^ and z' we obtain the connective determinant 

<^nl-fl, 4>m-m, J^m-i. 
Jn-e, 7n-u, Gn-i 

Vp-e, Vp-a, -ffjj-i 

which will be of the degree 

m + n + p — (6 + a + i), 

that ia of the degree (n+p- l-1) in x and y; and the number of such 
connectives by principle (c) is p. 

Again, by augmentation we can raise each of the functions U, V, W to 
the same degree as the connectives, and by principle (6) the number of such 
will be 

n + p — ra—i, 

p-h 
n— I, 
from U, V, W respectively, together making up the number 
2n + 2p -m-3t. 

Hence in all we have 2n + 'Ip — 3t equations ; and the number of terms 
to be eliminated will be, n+p — i arising from F,n, (r„, Hp, and n+p—2i 
from -Fm-i, Gn-i.> Sp_^; together making up the proper number 2n+2p — Si. 

Each connective contains ternary combinations of the coefficients, namely 
one of the coefficients belonging to that part of U, V, W which contains z\ 
and two coefiicients from the other part : the dimensions of the resultant in 
respect of the coefficients of the former will hence be readily seen to be equal 
to the number of connectives + the number of terms in the augmentatives 
into which z'- enters, that is, will equal m-'rn+p — 2i; the total dimensions 
of the resultant in respect to all the coefficients of U, V, W will be 

Sm + {2n + 2p —m — Si), 

that is, 2m + 2n + 2p — Si,; 

and consequently, in respect to the coefficients of i^,„; (?„ ; Hp, will be of 

(2m + 2n+2p-Si)-{m+n + p- 2t), 

that is, oim + n+p—i dimensions. This result, which is of considerable 
importance, may be generalized as follows. 

Returning to the general system (A), for which we have proved that the 
total dimensions of the resultant are 

(r-l)(m + n+...+p)-r(t + f' + ...+(t)). 



40] the Dialytic Method of Elimination. 261 

let the coefficients of the column of partial functions 

be called the first set ; the coefficients of the column 

Fm-L, 



the second set, and so forth ; then the dimensions in respect of the 1st, 
2nd ... (r— l)th sets respectively are s, s — t, s — l... s — {i), where 

s = m+n + &c. -\-'p— {i+ I + &c. + (t)). 

The important observation remains to be made, that all the above results 
remain good although any one or more of the indices of dimension of the 
partial functions in the system (A), as m — t, m — t', h — t, &c., should become 
negative, provided that the terms in which such negative indices occur be 
taken zero, as will be apparent on reviewing the processes already indicated 
upon this supposition. If we take 

m = n = ... =p, and t= t' = &c. =(t) = m — e, 

the exponent of the total dimensions of the resultant becomes 

(r — 1 ) rm — r{i — 2) (m — e) 

= rm + r (t — 2) e, 

when 6 = 0, this becomes mr, which is made up of 2m units of dimension 
belonging to the coefficients of the first column, and of m belonging to each of 
the (r — 2) remaining columns. Consequently, if we have 

Fm{oo, y)+^\-\-^'X' = Q, 

Gm (x, y) + 7JX + ri'\' = 0, 

Hr^ix, y)+^X + ^'X'=0, 

K^{x, y)+ex + 0'X'=O, 

or any other number of equations similarly formed, the result of the 
elimination is always of m dimensions only in respect of ^, t), f, 0, or of 
^', 7]', t,' , & , and of 1m in respect of the coefficients in F, G, H, K. 

I now proceed to state and to explain some seeming paradoxes connected 
with the degree of the resultant of such systems of defective functions as 
have been previously treated of in this memoir, as compared with the degree 



262 On Extensions of [40 

of the general resultant of a corresponding system of complete functions of the 
same number of variables. 

In order to fix our ideas, let us take a system of only three equations of 
the form 

F„ {x, y) + F,,^, {x, y) z^ = 0\ 
Gn{x,y)+G^Sx,y)z^ = o\. (B) 

Hj,{x, y) + Hp_,(x, y)z'- = 0} 

The resultant of this system found by the preceding method is in all of 
2m + 2n + 2p — Si dimensions. But in general, the resultant of three 
equations of the degrees m, n, p is of 7nn + mp + np dimensions. 

Now in order to reason firmly and validly upon the doctrine of elimination, 
nothing is so necessary as to have a clear and pi'ecise notion, never to be let 
go from the mind's grasp, of the proposition that every system of n homo- 
geneous functions of n variables has a single and invariable Resultant. 
The meaning of this proposition is, that a function of the coefficients of the 
given functions can be found, such that, ivhenever it becomes zero, and 
never except when it becomes zero, the functions may be simultaneously 
made zero for some certain system of ratios between the variables. The 
function so found, which is sufiScient and necessary to condition the possibility 
of the coexistence of the equality to zero of each of the given functions, is 
their resultant, and by analogy they may be termed its components. It 
follows that if J? be a resultant of a given system of functions, any numerical 
multiple of any power of R or of any root of R when (upon certain relations 
being supposed to be instituted between the coefficients of its components) 
R breaks up into equal factors, will also be a resultant. This is just what 
happens in system (B) when m = n = p = t; the resultant found by the 
method in the text is of the degree 3m ; the general resultant of the system 
of three equations to which it belongs is of the degree 3m^ ; the fact being, 
that the latter resultant becomes a perfect ?ftth power for the particular 
values of the coefficients which cause its components to take the form of the 
functions in system (B). 

Suppose, however, that we have still m = n=p, but t less than in, 
6m — Si will express the degree of the resultant of system (B) ; but this is 
no longer in general an aliquot part of 3m-, and consequently the resultant 
of system (B) that we have found is no longer capable in general of being 
a root of the general resultant. The truth is, that on this supposition the 
general resultant is zero ; as it evidently should be, because the values 

- = 0, - = satisfy the equations in system (B), except for the case of m = 4 ; 

consequently the resultant furnished in the text, although found by the same 
process, is something of a different nature from an ordinary resultant ; it 



40] the Dialytic Method of Elimination. 263 

expresses, not that the system of equations (B) may be capable of coexisting, 

but that they may be capable of coexisting for values of - , - other than 

and 0. This is what I have elsewhere termed a sub-resultant. But there 
is yet a further case, to which neither of the above considerations will apply. 
This is when m, n, p are not equal, but p — i, = Q. 

On this supposition the degree of the resultant of (B) becomes 2m+1n—p, 
which ia general will not be a factor of mn + mp + np ; and in this case it 

will no longer be true that the values - = 0, - = will satisfy the system (B), 

inasmuch as the last equation therein cannot so be satisfied. Now, calling 
the general resultant R and the particular resultant R', if R' should 
break up into factors so as to become equal to {r'Y x (s')' ... (<')", it might be 
the case that R should equal {r'Y . {s'Y ... (t'Y, and there would be nothing in 
this fact which would be inconsistent with the theory of the resultant as 
above set forth ; but suppose that R' is indecomposable into factors, then 
it is evident that we must have R = R' . R", and consequently that the 
existence of such a particular resultant as R' will argue the necessity of 
the existence of another resultant R" ; in other words, the resultant so 
found cannot be in a strict sense the true and complete resultant for the 
particular case assumed, and yet the process employed appears to give the 
complete resultant, or at least it is difficult to see how the wanting factor 
escapes detection. To make this matter more clear, take a particular and 
a very simple case, where m = 2, n = 2, ^ = i = 1, so as to form the system of 
equations 

Am? + Bxy + Gy'' + {Dx + Ey) z=Q-\ 

A'a^+B'xy+Gy+{D'cc+E'y)z=o\. (C) 

loo + my +nz = OJ 

By virtue of my theorem, the degree of the resultant R' is 

2 (2 + 2 + 1)- 3. 1 = 7, 

but the resultant R of the system 

Ax- + Bxy + Gy"- + {Dx +Ey)z +Fz"- = Ol 
^ V+ B'xy+ G'y^+ {D'x+ E'y) z+F'z" = q\, (D) 

Ix + 'my +nz = OJ 

which becomes identical with the former when i^ = 0, i'" = is of 
2x 2 + 2 X 1 + 2 X 1, 

that is, of 8 dimensions. Hence it is evident that when F = 0,F' = 0, R must 
become R' x R". 



264 Extensions of the Dialytic Method of Elimination. [40 

It will be found in fact*, that on the supposition of ^=0, F'=0, R becomes 
equal to N x R' ; and accordingly, besides the portion R of the resultant 
of system (C), found by the method in the text, there is another portion 
N which has dropped through ; but it may be asked, is N truly a relevant 
factor? were it not so, the theory of the resultant would be completely 
invalidated; but in truth it is; for iV"=0 will make the equations in 
system (C), considered as a particular case of system (D), capable of co- 
existing; the peculiarity, which at first sight prevents this from being 

obvious, consisting in the fact that the values of -, ^ which satisfy the 

z z 
three equations when iV = become infinite. 

Thus, finally, we have arrived at a clear and complete view of the relation 
of the particular to the general resultant. 

The general resultant may be zero, in which case the particular resultant 
is something altogether different from an ordinary resultant ; or the particular 
resultant may be a root of the general resultant, or it may be more generally 
the product of powers of the simple factors, which enter into the composition 
of the general resultant; or lastly, it may be an incomplete resultant, the 
factors wanting to make it complete being such as when equated to zero, will 
enable the components of the resultant to coexist, but not for other than 
infinite values of certain of the ratios existing between the variables. 

Without for the present further enlarging on the hitherto unexplored and 
highly interesting theory of Particular Resultants, I will content myself 
with stating one beautiful and general theorem relating to them; to wit, 
"if F=0, G=0, &c. be a given system of equations with the coefficients left 
general, and R be the resultant of F, G, &c., and if now the coefficients in 
F, G be so taken that R comes to contain as a factor or be coincident with 
R'"^, then will jR' = indicate that (when the coefficients are so taken as 
above supposed) F= 0, G=0, &c. will be capable of being satisfied, not, as in 
general, by one only, but by m distinct systems of values of the variables 
in F, G, &c., subject of course to the possibility, in special cases, of certain of 
the systems becoming multiple coincident systems." 

I pass on now* to the more recondite and interesting theory of the 
resultant of Ternary Aggregative Functions, that is to say, functions of 
the form 

Fm(x, y, z)+F,n-,{x, y, ^)!!' + &c. ... +i^„,_„ («, y, z)t^\ 

which will be seen to admit of some remarkable applications to the theory 
of reciprocal polars. 

[* See the Author's remarks below, p. 283.] 



41. 



ON A REMAKKABLE DISCOVERY IN THE THEORY OF 
CANONICAL FORMS AND OF HYPERDETERMINANTS. 

[Philosophical Magazine, II. (1851), pp. 391 — 410.] 

In a recently printed continuation* of a paper which appeared in the 
Cambridge and Dublin Mathematical Journal, I published a complete 
solution of the following problem. A homogeneous function of x, y of 
the degree 2n + l being given, required to represent it as the sum of w + 1 
powers of linear functions of x, y. I shall prepare the way for the more 
remarkable investigations which form the proper object of this paper, by 
giving a new and more simple solution of this linear transformation. 

Let the given function be 

ao«™+' + {In + 1) a^x-^y + |(2re + 1) (2w) a„x'"''-^y- + ... + a,„+i2/""+S 

and suppose that this is identical with 

{p^x + q^y)'^+^ + (p^x + q^yT'^^ + &c. + {pn+^x + qn+,y)-"+K 

The problem is evidently possible and definite, there being 2w + 2 
equations to be satisfied, and (2?i + 2) quantities p^, q^, &c. for satisfying 
the same. 

In order to effect the solution, let 

q._=p,X,, 

&c. = &c. 

9'jl+l = Pn+l\l+l, 
[* p. 203 above.] 



266 On a remarhaMe Discovery in the Theory of [41 



we have then 






p. 



+ ...+_p:,+Y =a„, 



p^:^+iT,n +p^^^+lXJ^ + . . . + p::;t'^'Wi = «»> 

Eliminate p^, p^ ... pn+i between the 1st, 2nd, 3rd ... (?i + 2)th equations, 
and it is easily seen that we obtain 

Again, eliminating in like manner pi™+^Xi, ^2^+^X2 ...^^^^X^+i between 
the 2nd, 3rd ... (n+ 3)th equations, we obtain 

(^n+2 ~ (^n+l ^^1 + ■ • • + OiXiXo . . . X^+l = " ) 

and proceeding in the same way until we come to the combination of the 
(w + l)th ... (2/1 + 2)th equations, and writing 

iXiX2 = §2, 



we find 



XjX, ... Xjj+j — Sn+1 , 

an+i -an Si + an-1 Sa . . . + a^Sn+i = 0, 
an+i - a„+iSj + an S^ ... + (hSn+i = 0, 

^11+3 — ^^71+2 Si + dn+l S2 . .. + £12^)1+1 = 0, 



^271+1 ^271 Sir ^271—1 S2 • • • "T dn Sji+i — " ■ 

Hence it is obvious that 

(x + \y) (x + X22/) ...(x + \n+i y) 
is a constant multiple of the determinant 

rt+1 ) ^n > ^71—1 • • • <^o 



fJ«+2i 



Ol 



^n+l J ^371 > ^271—1 ' • • ^71 

* These equations in their simplified fonn arise from the ordinary result of elimination, in 
this case containing as a factor the product of the differences of the quantities X^, \, ... X„^.i. 



41] Canonical Forms and of Hyperdeterminants. 267 



Hence Xj, X. ••• \i+i are known, and consequently 

Pi, Pi ■■■Pn+i, qi, ?2--- qn+i 
are known, by the solution of an equation of the {n + l)th degree. 
Thus suppose the given function to be 

F=ax''' + bha^y + IQcaPy'^ + lOdx'y^ + bexy^ + 10/i/= 
= ( Pia; + g, y)= + ( p^x + q^yj + ( ^3« + q%yf, 
we shall have, by an easy inference from what has preceded, 

{p^x + q^y) {p^x + q^y) {p^x + q^y) 
equal to a numerical multiple of the determinant 
a?, — a^y, xy"^, — y^ 

d, c, b, a 

e, d, c, h 

f, e, d, c 
The solution of the problem given by me in the paper before alluded to 

presents itself under an apparently different and rather less simple form. 
Thus, in the case in question, we shall find according to that solution, 

{pix+q^y){p^x + q,y)(p,x + q,y) 
equal to a numerical multiple of the determinant 
ax + by, bx + cy, ex + dy 
bx + cy, ex + dy, dx+ ey 
ex + dy, dx + ey, ex + fy 

The two determinants, however, are in fact identical, as is easily verified, 
for the coefficients of a? and y^ are manifestly alike ; and the coefficient of x^y 
in the second form will be made up of the three determinants, 

a, b, d \ , I a, e, c 

b, c, e \ \ b, d, d 

I 

c, d, f \ \e, e, e 

of which the latter two vanish, and the first is identical with the coefficient 
oiafiy in the first solution. The same thing is obviously true in regard of the 
coefficients of xy- in the two forms, and a like method may be applied to 
show that in all cases the determinant above given is identical with the 
determinant of my former paper, namely 

afjX + aiy, a^x + a^y ... a„a; + a„+iy 

aiX+ a2y, a^x -'r a^y ... Un+iX + an+^y 



h. 


b, 


c 


c, 


c, 


d 


d, 


d, 


e 



a^x + a,i+iy, a„+i« + a^+iy . . . a.^iX + a^n+iy 



268 On a remarkable Discovery in the Theory of [41 

Thus, then, we see that for odd-degreed functions, the reduction to their 
canonical form of the sum of (w + 1) powers depends upon the solution of one 
single equation of the {n + l)th degree, and can never be effected in more 
than one way. 

This new form of the resolving determinant affords a beautiful criterion 
for a function of x, y of the degree 2w + 1 being composed of n instead of, 
as in general, («,+!) powers. In order that this may be the case, it is obvious 
that two conditions must be satisfied ; but I pointed out in my supple- 
mental paper on canonical forms, that all the coefficients of the resolving 
determinant must vanish, which appears to give far too many conditions. 
Thus, suppose we have 

ax' + "thx^y + llcaff + Zhdaf'y^ + S5ex>y* + 2lfxY + '^gxy'^ + %'■ 

The conditions of catalecticism, that is, of its being expressible under the 
form of the sum of three (instead of, as in general, four) seventh powers, 
requires that all the coefficients of the different powers of x and y must 
vanish in the determinant 





y\ 




y^x, y^x", — ya?, 


x' 


) 




a, b, c, d, 


e 






b, c, d, e, 


f 






c, d, e, f, 


9 






d, e, / g, 


h 




ds, we must have five determinants, 






a, 6, c, d 


, 


a, c, d, e 


, 


a, 


b, c, e 


b, c, d, e 




b, d, e, f 




h, 


% d, f 


c, d, e, f 




c, e, /, g 




c, 


i, e, g 


d, e, f, g 




d, f, g, h 




d, 


e, f, h 


a, b, d, e 


, 


b, c, d, e 


, 






b, c, e, f 




c, d, e, f 








c, d, f, g 




d, e, f, g 








d, e, g. 


h 




e, f, g, h 









all separately zero. But by my homaloidal law*, all these five equations 
amount onlyto (5 — 4) (5 — 3), that is, to 2. I may notice here, that a theorem 
substantially identical with this law, and another absolutely identical with 
the theorem of compound determinants given by me in this Magazine, and 
afterwards generalized in a paper also published-}- in this Magazine, entitled 



[* p. 150 above.] 



[t p. 241 above.] 



41] Canonical Forms and of Hyperdeterminants. 269 



" On the Relations between the Minor Determinants of Linearly Equivalent 
Quadratic Forms," have been subsequently published as original in a recent 
number of M. Liouville's journal. 

The general condition of mere singularity, as distinguished from cata- 
lecticism, that is, of the function of the degree In + 1, being incapable of 
being expressed as the sum of n + 1 powers, is that the resolving resultant 
shall have two equal roots; in other words, that its determinant shall be 
zero. 

Mr Cayley has pointed out to me a very elegant mode of identifying the 
two forms of the resolving resultant, which I have much pleasure in sub- 
joining. Take as the example a function of the fifth degree, we have by 
the multiplication of determinants. 



y\ 


-y^x 


yx\ 


-a? 




1, 


0, 


0, 





a, 


b, 


c, 


d 


X 


X, 


V' 


0, 





b, 


c, 


d, 


e 




0, 


X, 


y. 





c, 


d, 


e, 


f 




0, 


0, 


X, 


y 



2/', a, b, c 

0, ax + by, bx + cy, ex + dy 

0, bx + cy, ex + dy, dx + ey 

0, ex + dy, dx+ ey, ex +fy 

which dividing out each side of the equation by y'', immediately gives the 
identity required, and the method is obviously general. 

Turn we now to consider the mode of reducing a biquadratic function of 
two letters to its canonical form, videlieet 

{fa + OVY + e*^ + %)* + 'om {fa + gyf {hx + kyf. 

Let the given function be written 

ax^ + 4ibx^y + &cx^y'^ + idxy'' + ey': 

Let g=fk^, k = hXi, 'nip}C- = fi, Xi + \2 = Si, \{Ki = s^, 

then we have 

/<■ + A^ + 6/i = a, 

4/^Xi + 4/i^X, + <6fi (2si) = 46, 

6/'Xi= + i6V\i + 6/i (si= + 2s0 = 6c, 

4/^X,' + 4A*X/ + 6/A i^s^si) = ^d, 

faXi'+h'\,^+6fis,' = e. 



270 On a remarTcdble Discovery in the Theory of [41 



Eliminating/ and h between the first, second and third; the second, third 
and fourth; and the third, fourth and fifth equations successively, we obtain 

as^—bsi+c — fi (8S2 — 2si') = 0, 

bs2 — csi + d — fi (4tSiSi— Si^) =0, 

cs2 — dsi+e—fi (Sso^ — 2si^s„) = 0. 

{2s,- — 8S2) /"• = !', 

aSa — bsi + (c+v)=0, 



^)s,+d = 0, 



Let now 

and we shall have 



(c + v)s2 — dsi+e = 0. 
Hence v will be found from the cubic equation 
a, b, c + v 



that is, 



26, 2c - V, 
c + v, d, 



v^-v {ae - ^bd + 3c=) + 



U 



= 0, 



a, 


b, c 


b, 


c, d 


c, 


d, e 



= 0, 



in which equation it will not fail to be noticed that the coefficient of v- is 
zero, and the remaining coefficients are the two well-known hyperdeter- 
minants, or, as I propose henceforth to call them, the two Invariants of 
the form 

ax'^ + 'kbafiy + Qcx^y- + idosy^ + ey* ; 

be it also further remarked that 

v = 8 {{Si^ — S2) fi, 
in which equation the coefficient of 8fi is the Determinant or Invariant of 

aP + Sixy + s^y". 

When V is thus found, Sj, S2, and fi, being given by the equations in terms of v, 
are known, and by the solution of a quadratic \, Xo become known in terms 
of Si, S2, and /, h in terms of Xj, X^, fi, and the problem is completely deter- 
mined. The most symmetrical mode of stating this method of solution is 
to suppose the given function thrown under the form 

{fa + gyf + (/i^' + g^y)' + 6e (/« + gyf {f^x + g.yf. 
Then writing 

O + gy) {fi«= +giy) = Lx" + Mxy + Ny-, 



41] Canonical Forms and of Hyperdeterminants. 271 

- V, the quantity to be found by the solution of the cubic last given, becomes 

I shall now proceed to apply the same mettiod to the reduction of the 
function 

a„«8 + ^a.x'y + 2Sa^xY + Se^j^y + 70a,xY + oQa,a?f 

+ 28a,xy + 8arxf^rasy', 
under the form of 

(p,x + q,yy + {p^x + r^',yy + {p,x + q,yf + {p,x + q,yf 

^J__.^±.l"Oe (p^x + q^yf{p^x + qoyf (p^x + q,yf (p,x + q^yf. 
It will bj^'5'nvenient to begin, as in the last case, by taking 

-^ qi=Pi\, qi^Pi'^, qs=ps'>^s, qi=Pi'^i, 

epiPiPiPi = m, 
and 
{x + Xit/) {x + X22/) {x + Xs^) {x + X^y) = x^ + SiOify + s^a?y''- + s^xy'^ + 843/* = JJ, 

we shall then have nine equations for determining the nine unknown 
quantities of the general form 

Px^\ ■^pi'l^i +^3%' + l^/X/ + M^m = a„ 

where t has all values from to 8 inclusive, and where 

1.2....1.2. (8-.) 
1.2... 8 

multiplied into the coefficient of y'-a?~'- in U'^. 

Taking these nine equations in consecutive fives, beginning with the first, 
second, third, fourth, fifth, and ending with the fifth, sixth, seventh, eighth, 
ninth, we obtain the five equations following: — 

a,iSi — ajSj + ttaSa — a^Si + ajSo — mN-^ = 0, 

(Z1S4 — OaSs + tt'sSa — ^i^i + (^e^o — m.N'i — 0, 

a^Si — a^s-i + a^s^ — a^Si ■+ a^So — mNs = 0, 
a^Si — a^Sa + ajSa — cieSi + ctySo — inNi = 0, 
aiSi — a^Ss + agSa — a,Si + agSo — mN^ — 0, 



where 



iVi = ilf„S4 - ifiSs + M„_s^ - M,Si + Mi, 
M^ = M^Si - M^s, + M^So. - M,s, + M„ 
Ns = 71/284 - M^S:, + MiS^ — M^St, + Mg, 
Ni = M.,Si - M^s^ + ilfsSa - M^s-, + Mj, 
N, = M,Si - M,s, + M,s^ - M,s^ + M^. 



272 On a r<"markahle Discovery in the Theory of [41 



Developing now V", we have 

5 5 5 

i^/4=2s4+2SlS3 + S2^ if3 = 2SiS4 + 2«2S3, ilfs = 5S2S4 + 2 S3', 

35 



if, = — 8354, M8 = 70s4^ 



Hence 



9 3 

iVj = I8S1S4 - 2 Si'Ss + 2 SiSj', 
iVs = I2S2S4— SSiSoSj + Sa", 

9 3 

iVi = I8S3S4- 2 S1S3' + 2 *2' ^3, 

iVs = 72S4= - ISSjSaSj + <OSiSi. 

Hence we have 

N^=1'2I, iV"„=72/^, iV3 = 72/^!, N, = 12I^, N', = 72Is„ 
4 o 4 

where it will be observed that / is the quadratic invariant of U. 

Making now 

72m/ = V, 

we shall have the five following equations : — 

a„S4— ai53+ a.s, — a^Si 



OiSj — a.2S3+ tts 



a^Si — a,S3 + [ a^ — 5 Sa — a^Si 



-{ai+'-x] s, + a, =0, 

+ tte =0, 



0354— (a4 + J ISs + asSa— «6Si 



+ av 



= 0, 



(a4 — i/) S4 + a^Ss — agS, — o-Si +ag = ; 

so that the problem reduces itself to finding v, which is found from the 
equation of the fifth degree: — 

tto. 0,1, (h, O-a, Ui—V 



an. 


a„ 




a4+^, 


a, 


0.3, 


a4- 


V 

6' 


a„ 


as 


V 


an, 




a„ 


Oj 


(^5, 


<^a, 




<h, 


as 



/ 



41] Canonical Forms and of Hyperdeterminants. 273 

V, it will be observed, being 72 times the quadratic invariant of 

{p^x + q,y) {pr.x + q„y) (p,x + q,y){piX + q^y), 
the function being supposed to be thrown under the form of 

t{pix + q,yf + 706 (pyV + q,yf(p.,x + q„y)-{p^x + q^yJip^x + q^yf. 
It is obvious that in the equation for finding v, all the coefficients being 
functions of the invariable quantities p^, q^, &c., and e, must be themselves 
invariants of the given function ; so that the determinant last given will 
present under one point of view four out of the six invariants belonging 
to a function of the eighth degree, and these four will be of the degrees 
2, 3, 4, 5 respectively*. 

I shall now proceed to generalize this remarkable law, and to demonstrate 
the existence and mode of finding 2n consecutively-degreed independent 
invariants of any homogeneous function of the degree 4n., and of n+1 con- 
secutively-even-degreed independent invariants of any homogeneous function 
of the degree 4n + 2 ; a result, whether we look to the fact of such invariants 
existing, or to the simplicity of the formula for obtaining them, equally 
unexpected and important, and tending to clear up some of the most obscure, 
and at the same time interesting points in this great theory of algebraical 
transformations. 

In the first place, let me recall to my readers in the simplest form what is 
meant by an invariant f of a homogeneous function, say of two variables 
X and y. If the coefficients of the function f{x, y) be called a, b, c ... I, and 
if when for x we put Ix + my, and for y, nx+py, where Ip— 77111=!, the 
coefficients of the corresponding terms become a', b' ... I'; and if 

I(a,b...l) = I{a',b'...l'), 
then /is defined to be an invariant of/ 

Let now f{x, y) be a homogeneous function in x, y of the 2(.th degree, 
and write 

'Tx^-^T^f^^^^y^+^^'n^-^yy^P' 
i^dx'^'^dy)-^^^'"^ ™^' ""^ +py) + '^ ('?* - ^yy = -f"' 

where ^ and ?/ are independent of x, y, and Ip — m?i = 1. 

Let x' = lx + my, 

y' = nx + py, 

,, J, cZ ^ _ £ '^*' ^ fc "^2/' ^ '^*' '^ dy' ^ 

dx dy dx dad ^ dx dy dy dx' dy dy' ' 

* The reasoning in this paragraph seems of doubtful conclusiveness. It may be accepted, 
however, as a fact of observation confirmed and generalized by the subsequent theorem, that the 
coefficients are invariants. 

t Olim, Hyperdeterminant, Constant derivative, 
s. 18 



274 On a remarhable Discovery in the Theory of [41 

and if we now write 

l^ + mrj = ^', 

Again, from the equations between x', y , x, y, we find 

vx' — my' , , 

X = ±— ^ — =px — my , 



pi- 



mn 



ly — nx , , , 

y—-^ = tv —nx ; 

"^ pL — mn ^ 

therefore 7]X — ^y=(p'r] + n^) x — {m-rj + l^) y' = 7]'x' — ^'y'. 

Hence P' = (^f — + 7,' ^-, j / {x\ y') + X (77V - ^'yj. 

. . d , d d 



d _ d d 

drj d^' drj' ' 



Hence 






d^J dr) 



But P' being of t dimensions in ^' and 97', and also in x and 2/. each 
of the equations above written will be of t dimensions in x and y, and of no 
dimensions in f ', 7/' ; in fact, the successive terms of the right-hand members 
of the above t + 1 equations will be multiples of the {i + 1) quantities 

{x')\ (xJ-Hj', (xy--y''- ...(yj. 

Consequently a linear resultant may be taken of 

treating «'', x'-~^y' ... y'' as independent, and as quantities to be eliminated; 
and this, according to a well-known principle of elimination, will prove 



41] Canonical Forms and of Hyper determinants. 275 



the linear resultant of the foregoing equations to be equal to the linear 
resultant of 

d \-^ d ^, / d 



\d^'J ' [d^'j dr, \d7,'/ 



multiplied by the determinant 

Z', fwZ'~^, ^i(i—l)n'-l'~-, 



m', 



I' ^p, ^i{i — l)m'p' ", ..., 



p. 



This last written determinant may be shown from the method of 

its formation to be equal to {Ip — mn) ^ , that is, to unity, because 
Ip — mn — 1. Again, since 



x'' = I'x"- + d'-~^'maf ^y + &c. + m'-y\ 
c''~'2/' = l'-~^nx'-+ (l'-~^n + (s — 1) l'-~^mn) x'-~^y + &c. + ■; 



-'py-, 



y'<- = n'-x'- + iiV-~^px'-~'^y + . . . + p'-y'-, 

the resultant of (-7^) P ... (-j- J P', obtained by treating «', x'-'~'^y ... 3/' as the 

eliminables, will be equal to the resultant of the same functions when 
x'\ a;''"~^2/' .•• 2/' ^'^^ taken as the eliminables* multiplied by a power of the 
determinant 

l\ .... ??!■' 



m' ^p 



which determinant, like the last, is unity. Thus, then, we have succeeded 
in showing that the resultant obtained by eliminating «', x'~^y ... y- 
between 



©>■ 



'""■|^'-(|)> 



is equal to the resultant obtained by eliminating («')', x''-~^y' ...y'' between 

fdy fdy-^d^p, (d\p,, 

* For the statement of the general principle of the change of the variables of elimination, 
see my paper in the March Number, 1851, of the Camb. and Dub. Math. Jour. [p. 186 above]. 

18—2 



276 On a remarkable Discovery in the Theory of [41 

or, which is evidently the same thing, the resultant obtained by eliminating 
x\ x'-~'^y ... y^ between 

d^J ' \d^J dv \dv) ' 

that is to say, this last resultant remains absolutely unaltered in value when 
for x, y we write respectively 

Ix + my, 

nx + py, 
provided that Ip — mn = 1. 

Hence by definition this resultant is an invariant f{x, y), and A, being 
arbitrary, all the separate coefficients of the powers of X in this resultant 
must also be invariants. I proceed to express this resultant in terms of 
X and the coefficients of (x_, y). Let ot = 1 . 2 . 3 . . . t and 

. Mrf)"^-(£)>^(-'>' 

1 f dy-'' d „ f dv--" d 



1 fd 



and 



(|)-(0-=©'""(|)V-<-,y-=.=^., 



1)'^ =(!)>-- 



f{x, y) = »„«" + "^ia^x^'-Hj + 1 (2t) (2t - 1) a^oe^'^'y'' + &c. + a.^y'-K 
We find, writing o-X for X, where cr = 2i(24— 1) ... (t + 1), 

- ^i = aoa;'+ tai«'~'2/+ \i{i,— V) a^x'-~'^y- ... 

+ I i (t - 1) a.-^x^y--"^ + M.-^xy--" + a.y- + X (- yY, 

— E^ = a^x'- + ia2x'-~^y+ ^(.(i— l)a3«'~^2/^... 

+ i 1 (t - 1) a.^ix^y-'- + ia.xy--^ + a,+i2/' + X (- yy-^'x, 

1 r. 

+ 1 1 (t - 1) a.x'y'--' + M.+ixy--"- + a.+^y- + X (- 3/)'-^a;=, 



— j&j+i = a^a;' + &c. + Xa;' 



41] Canonical Forms and of Hyperdeterminants. 

accordingly, by eliminating 

. x'-, LX'-^y, ^t(t — 1) a;'~^2/^ ... 2/', 
we obtain as the required resultant*, 

di+i, ai + -, cii_i, ... (h 

X 



277 



<^l+2) <^c+l) 



a, + 



■'('-1)' 



ci( + X, 



Inasmuch as all the coefiScients of X in this expression are invariants of 
f{x, y), and there are no invariants of the first order, it is clear that the 
coefficient of X' must be always zero, which is easily verified. 

Again, if i is odd, the determinant remains unaltered if we write —X for 
X ; hence when f(x, y) is of the degree 4e + 2, all the coefiScients of the odd 
powers of X disappear. Thus, then, our theorem at once demonstrates that a 
function of x, y of the degree 46 has 2e invariants of all degrees from 
2 up to 2e + 1 inclusive, and that a function of x, y of the degree 46 + 2 
has 6 + 1 invariants whose degrees correspond to all the even numbers in the 
series from 2 to 2e + 2. 

But in order that the proposition, as above stated, may be understood in 
its full import and value, it is necessary to show that these invariants are 
independent of one another, which is usually a most troublesome and difficult 
task in inquiries of this description, but which the peculiar form of our 
grand determinant enables us to accomplish with extraordinary facility. In 
order to make the spirit of the demonstration more apparent, take the case 
of a function of the twelfth degree, whose coefficients, divided by the 

12 11 

successive binomial numbers 1, 12, — ^ — , &c. may be called 

a, h, c, d, e, f, g, h, i, j, k, I, m. 



* Mr Cayley has made the valuable observation, that X (given by equating to zero the above 
determinant) may be defined by means of the equation 



(^i-i,4y{^(^''-')><*(f''')>=^^(^'^)' 



^ being itself a certain rational integral form of a function of the ith degree, the ratio of whose 
coefficients would be given by virtue of the above equations as functions of X and the coefficients 
olf(x,y). 



278 On a remarkable Discovery in the Theory of [41 



Our grand determinant then takes the form 

g -\-\ f, e, d, c, b, a 



I., .-i. y. 



d, c, h 



i h, 9 + Y^> f, e. ^. c 
i. *. ^. 5' -on' /' ^' '^ 



^, j. i, 



20' 



^. S' + ig- / 



h, 9-Q, f 



I, k, j, i, 

m, I, k, j, i, h, g + X 
Here it will be observed that 

a and m appear only 1 time. 



b and I 


2 times 


c and k 


3 ... 


d and j 


4 ... 


e and i 


5 ... 


f and h 


6 ... 


a 


7 ... 



Let now the coefficients be called 

H„ H„ H„ H„ H„ H„ 

H^ and H^ manifestly are independent. 

Again, if possible, let Ht =pH.?, then a and m would appear twice in H^, 
contrary to the rule. 

Hence Hi is independent of H^, H^. 

For a similar reason H^ cannot depend on H^, H^. 

Again, if possible, let 

H, =pHi + qH.Hi + rHi, 
Hi will contain b'^l'^, which by the rule cannot appear in H^H^ or in H^. 
Hence p = 0. 
Also Ha will contain hH"^ x the coefficient of V in 



.^ + f5)(^-^o)(^+r5)' 



41] Canonical Forms and of Hyjper determinants. 279 

which is not zero. And H^ also contains hi ; hence JI^B.^ will contain TfP. 
But Hi will evidently not contain 6' or P, or hH or W, nor can H^ contain 6'^^ ; 
hence gi = 0. Finally, Hi will contain c* and A*, but H^, can only contain as 
to these letters the combination &l?\ hence 7- = 0. 

Consequently H^ does not depend on H.^, H^, H^. As regards H^, H^, 
Hi, H^, H^ not vanishing, this may be made at once apparent by making 
all the letters but g vanish ; the -ff's then become identical with the 
coefficients of 

none of which are zero except that of X". The same or a similar demonstra- 
tion may be extended to H., and easily generalized ; hence, then, this most 
unexpected and surprising law is fully made out*. 

To return to the subject of canonical forms, I have not found the method 
so signally successful in its application to the 4th and 8th degrees, conduct to 
the solution of other degrees, such as the 6th, 12th, or 16th, of all of which 
I have made trial ; possibly another canonical form must be substituted to 
meet the exigency of these casesi* ; and it may be remarked in general, that 
if we have a function of the (2n.)th degree, the canonical form assumed 
may be taken, 

where V, in lieu of being the squared product of 

(p,x + q,y), (p„x + q„y), ..., (p^x + q„y), 

* This demonstration, however, does not extend to show that the coefficients of the powers of 
X may not possibly be dependents, that is, explicit functions of one another combined with other 
invariants not included among their number, or of these latter alone. For example, in the case 
of the 12th degree, we know by Mr Cayley's law that there must be two invariants of the 
4th order. Our determinant gives only one of these. Call the other one K^; by the above 
reasoning it is not disproved but that we may have 

H^=pH^ + qH^Hi + rH^^ + sHJC^ . 

I believe, however, that the ff's may be demonstrated without much difficulty to be primitive 
or fundamental invariants. The law of Mr Cayley here adverted to admits of being stated in the 
following terms : — The number of independent invariants of the 4th order belonging to a 
function of x, y of the ?ith degree is equal to the number of solutions in integers (not less than 
zero) of the equation 2x + 3y = n-S. Vide his memorable paper (in which several numerical 
errors occur against which the reader should be cautioned) "On Linear Transformations," vol. i. 
Gamhridge and Dublin Mathematical Journal, new series. There is no great difficulty in showing, 
by aid of the doctrine of symmetrical functions, that there can never be more than one quadratic or 
one cubic invariant, and in what cases there is one or the other, or each, to any given function 
of two variables. The genera] law, however, for the number of invariants of any order other 
than 2, 3, 4 remains to be made out, and is a great desideratum in the theory of linear trans- 
formations. 

t See the Postscript [p. 283] for a verification of this conjecture. 



280 On a remarkable Discovery in the Theory of [41 

may be any hyperdeterminant, or (as I shall in future call such functions) 
covariant of this product, understanding P {x, y) to be a covariant of 
f{x, y) when P (Ix + my, nx + py) stands in precisely the same relation to 
f(lx + my, nx + py) as P(x, y) to f{x, y), provided only that lp — mn=\. 
For the relation and distinction between covariants and contra variants, see 
a short article of mine* in the Cambridge and Dublin Mathematical Journal 
for this month. In endeavouring to apply the method of the text to the 
Sextic Function 

aa^ 4- &)afiy + Xbcafy'^ + 20dofy'' + 1 5ex^y* + 6fxy^ + gy^, 

thrown under the form 

l<(px+qyy + 20eU^ 
where 

U={piX + qiy) {piX + q^y) (p^x + q^y) = s^a? + s-,a?y + s^xy'^ + S3y\ 
I obtain the following equations : 

aSs — bsr. + csi — dso=e {lQ2so% — o4soSiS2 + 12si^), 
bss — cs^ + dsi — eso = 6 (54S|,SiS3 + 6si% — SGsoSa^), 
CS3 — ds2 + esi —fso = e (— 54S0S2S3 — QsiS^^ + SGsaSi"), 
dss - es2 +fsi-gso = e (- 162soS3= + B'tsiS^Ss 4- 12s/). 

In these equations, if we call the quantities multiplied by e respectively 
L, M, If, P, we shall find 

S3L - g s^ilf - g SiiV + SoP = 0, 

and SsL - S.2M - s^N + SoP = I ; 

where / denotes the determinant, or, as I shall in future call such function 
(in order to avoid the obscurity and confusion arising from employing the 
same word in two different senses), the Discriminant"!", which is the biquadratic 
(and of course sole) invariant of the cubic function 

Soce^ + SiX-y + s.2xy- + s^yK 

The reduction of the function of the fourth degree to its canonical form 
may be effected very easily by means of the properties of the invariants of 

[* p. 200 above.] 

+ "Discriminant," because it affords the discrimen or test for ascertaining whether or not 
equal factors enter into a function of two variables, or more generally of the existence or other- 
wise of multiple points in the locus represented or characterized by any algebraical function, the 
most obvious and first observed species of singularity in such function or locus. Progress 
in these researches is impossible without the aid of clear expression ; and the first condition of a 
good nomenclature is that different things shall be called by different names. The innovations 
in mathematical language here and elsewhere (not without high sanction) introduced by the 
author, have been never adopted except under actual experience of the embarrassment arising 
from the want of them, and will require no vindication to those who have reached that point 
where the necessity of some such additions becomes felt. 



41] Canonical Forms and of Hyperdeterminants. 281 

the canonical form, as I have shown in the Cambridge and Dublin Mathe- 
matical Journal. Accordingly I have endeavoured to ascertain whether 
the reduction of the sixth degi'ee might not be effected by a similar 
method. 

If we start with the form ax^-\-by^+ cz^ + QOma?y''z"', where x + y +z = {), 
which is only another mode of representing the canonical form previously 
given, we shall find that there are four independent invariants, of the second, 
fourth, sixth and tenth degrees. Calling these H^, H^, H^, Hj^, and writing 
Si, 52, S3 for a+b + c, ab + ac + bc, abc it will be found, after performing 
some extremely elaborate computations, that 

H^ = s^- 270m=, 

Hi = 6ms3 + 4,5^1% + 2lQm% + 891m^ 

ifg = 4s3= + 120s.,Ssm - {684s/ + 432S1S3} m^ 

+ (13 . 27 . 64s3 - 64 . SlSiS^) jn' + S.Sl. 169s.m* 
+ 7 . 128 . 729sim= + 16 . 729 . 239m''. 

Hio is too enormously Jong to attempt to compute ; but we can easily 
prove its independent existence by making m = 0, in which case the (deter- 
minant, or, to use the new term proposed, the) discriminant of a*" + by^ + cz^ 
becomes the product of the twenty-five forms of the expression 

(ab)i + {acf . 1* + (bc)i . 1* *. 

Now in general the value of such a product for a^+/3*. 1^ -f- 7^. 1° is obviously 
of the form 

(a+ /3 + 7)^ + a/37 {/(« + /3 + yy+9(a^ + olj + ^y)} ; 
for when a = or j8 = or 7 = 0, the product must become respectively 
(j3 + yY, (7 -1- a)^ and (a + /3)°. Moreover, without caring to calculate /, gf, it is 
enough for our present purpose to satisfj' ourselves that g cannot be zero, as 
then the product would have a factor (a + ^ + 7)^. Hence, then, on putting 

* Such a product in the language of the most modern continental analysis is, I believe, 
termed a Norm. If we suppose the general function of x, y of the 4th degree thrown under the 
form Au^ + Bv^+Cw*, where k + i! + w = 0, and the general function oix, y, z of the 3rd degree 
thrown under the form AiC^ + Bv^ + Cw^ + DS^, where 'ii + v + io + d=0, the theory of norms will 
afford an instantaneous and, so to speak, intuitive demonstration of the respective related 
theorems, and the discriminant (aliter determinant) of each such function is decomposable into 
the sum of a square and a cube. Each of these forms is indeterminate, in either case there 
being but two relations fixed between the coefBcients A, B, C; A, B, G, D; and we may easily 
estabHsh the following singular species o£ algebraical porism. In the first case 



and in the second case 



are invariable ratios. 
t/=-625, 5 = 3125. 



{ABCf:{AB + AC + BCf, 
(ABGDf : {2A'B^C^ - 2ABCDSABf 



282 On a remarJcable Discovery in the Theory of [41 

oi = hc, ^ = ac, 7 = ah, we see that the discriminant, when m is 0, will be of 
the form 

Ss'+fsi^s^'^ + gss^. 
But when m is 0, R^ vanishes, and there is no terra Sj or S3 in H2. Hence 
evidently the discriminant H^a just found cannot be dependent on H^, H^, 
or H^; nor is it possible to make 

H,, + pH,' + qH,'H„ 
that is, (p + 1) s./ +/s/s3= + gs-i^s^ 

a perfect square on account of g not vanishing ; so there is no H^ upon which 
ITio can depend. Hence, admitting, as there seems every reason to do, that 
the number of invariants of a function of x, y of the degree m is m — 2, 
we find that the four invariants in the case of the first degree are respectively 
of the second, fourth^ sixth, and tenth dimensions, a determination in 
itself, as a step to the completion of the theory of invariants, of no minor 
importance. 

But it seems hopeless by means of these forms to arrive at the desired 
canonical reduction. The forms, however, of H^, Hi, H^ are very remarkable 
as not rising above the first, first and second degrees respectively in s^, s^, S3. 
Also Hi vanishes when m = and H^ has been obtained by putting 

ax^ + hy^ + cz^ + QOmx-y'^z^ 
under the form of 

Ax^ + &Bx''y + loCxy + 20DxY + loExy + QFxif + Gy\ 

and taking the determinant 



A 


B 


C 


D 


B 


C 


D 


E 


C 


D 


E 


F 


D 


E 


F 


Q 



Consequently in general the vanishing of the above-written determinant will 
expi-ess the condition that a function of the sixth degree may be decomposable 
into three sixth powers. This also is true more generally. If F(x, y) be 
a function of 2i dimensions, the vanishing of the resultant in respect to 
x^, x^~'-y ... y^ (taken dialytically) of 

\dxj ' \dx) dy '" \dyj 
will indicate that F admits of being decomposed into i powers of linear 
functions of x, y*. 

In consequence of the greater interest, at least to the author, of the 
preceding investigations, I have delayed the insertion of the promised 
continuation of my paper on extensions of the dialytic method, which will 

* Sucb a function so decomposable may be termed meio-catalectic. Meio-catalecticism for 
even-degreed functions is the analogue of singularity for odd-degreed functions. 



41] Canonical Forms and of Hyper determinants. 283 

appear in a subsequent Number. I take this opportunity of correcting a 
trifling slip of the pen which occurs towards the end* of the paper alluded to. 

The values of - and - become zero, and not infinite, when iV"= 0; and the 

z z 

antepenultimate paragraph should end with the words " an incomplete 
resultant." The theorem also, in the last paragraph but one, should be 
stated more distinctly as subject to an important exception as follows. 

Whenever the resultant of a system of equations F = 0, G = 0, &c. 
contains a factor R'™, this will indicate that, on making R' = 0, the given 
system of equations will admit of being satisfied by m algebraically distinct 
systems of values of the variables, except in those cases where there is a 
singularity in the forms of F, G, &c., taken either separately, or in partial 
combination with one another. An example will serve to make the meaning 
of the exception apparent. Let F, G, H denote three quadratic equations 
in X and y, so that F=0, G = 0, H=0 maybe conceived as representing 
three conic sections. Let R be the resultant of F, G, H, and suppose the 
relations of the coefficients in F, G, H to be such that R = R"' ; then ii' = 
will imply the existence of one or the other of the three following conditions : 
namely, either that the three conies have a chord in common, which is the 
most general inference; or, which is less general, that two of the conies 
touch one another ; or, which is the most special case of all, that one of the 
conies is a pair of right lines. 

So, again, if we have two equations in x, and their resultant contains F^, 
this may arise either from one of the functions containing a square factor, 
or from their being susceptible, on instituting one further condition, namely 
of J^= 0, of having a quadratic factor in common between them. 

P.S. The conjecture made in the preceding pages has been since con- 
firmed by the discovery of a modification in the canonical form applicable 
to functions of the sixth degree, which simplifies the theory in a remarkable 
manner. Assume f{x, y), a function of the sixth degree, as equal to 

au' + hv^ + ciu^ + muvw (u — v)(v- w) {w — u), 
where u, v, w, linear functions of x and y, satisfy the equation 

u + V + w = 0; 
then will the product of uvw be capable of being determined by means of the 
solution of a quadratic equation, of the square root of whose roots the 
coefficients of uvw will be known linear functions. Thus by an affected 
quadratic, a pure quadratic, and a cubic equation, the values of u, v, w 
may be completely ascertained. The discussion of this theory, and of a 
general inverse method for assigning the true (in the sense of the most 
manageable) Canonical Form for functions of any even degree, will form 
the subject of a subsequent communication. 

[* p. 264 above.] 



42. 

ON THE PRINCIPLES OF THE CALCULUS OF FORMS. 
[Cambridge and Dublin Mathematical Journal, vii. (1852), pp. 52 — 97.] 

Part I. Generation of Forms*. 
Section I. On Simple Concomitance. 

The primary object of the Calculus of Forms is the determination of 
the properties of Rational Integral Homogeneous Functions or systems of 
functions : this is effected by means of transformation ; but to effect such 
transformation experience has shown that forms or form-systems must be 
contemplated not merely as they are in themselves, but with reference to 
the ensemble of forms capable of being derived from them, and which 
constitute as it were an unseen atmosphere around them. The first part of 
this essay will therefore be devoted to the theory of the external relations 
of forms or form-systems ; the second part to the analysis of forms : that is to 
say, the first part will treat of the Generation and affinities, and the second 
part of the Reduction and equivalences of forms. 

In its most crude and absolute, or, so to speak, archetypal condition a 
Rational Integral Homogeneous Function may be regarded as a linear 
function of several distinct and perfectly independent classes of variables. 

* It may be well at the outset to give notice to my readers of tbe exact meaning to be 
attached to the following terms : 

1. The linear-transformations are supposed to be always taken such that the modulus, 
that is, the determinant of the coefficients of transformation, is unity ; or, as it may be phrased, 
the transformations are uni-modular. 

2. The word Determinant is restricted in all cases to signify the alternate function formed in 
the usual manner from a group of quantities arranged in square order. 

3. The word Discriminant (typified by the prefix-symbol D ) is used to denote tbe deter- 
minant (usually but most perplexingly so called) of a homogeneous function of variables. 

4. The resultant of two or more homogeneous functions of as many variables is the left- 
hand side of the final equation (in its complete form and free from extraneous factors) which results 
from eliminating the variables between the equations obtained by making each of the functions 
zero. 



42] On the Principles of the Calculus of Forms. 



285 



The first step towards the limitation of this very general but necessary 
conception consists in imagining the total number of classes to become 
segregated into groups, and certain correspondences to obtain between 
the variables of a class in any group with some the variables in each other 
class of the same group. The investigations in this and the subsequent 
section will be confined exclusively to the theory of functions where the 
several classes of variables, if more than one, all belong to a single group, so 
that the variables in one class have each their respective correspondents 
in the remaining classes. Such a group may again be conceived to become 
subdivided into sets each of the same number of variables, and the corre- 
sponding variables in the different sets to become absolutely identical. This 
leads to the conception of a homogeneous function of related classes of 
variables of various degrees of exponency in respect to the several classes. 
The relation of the different classes, if containing the same number of 
variables (in which case the relation may be termed Simple) will be under- 
stood to be defined by their being simultaneously subject to similar or 
contrary operations of linear substitution; so that, for example, if x, y, z; 
^, 7), f are two such classes, when x, y, z are replaced by ax+hy + cz, 
a'x + h'y + c'z, a"x + h"y + c"z, respectively, ^, 77, f will be, acconiing to the 
species of the relation, subject to be at the same time replaced either by 
a^+hr] + c^, a'^ + h'rj + c'^, a"^ + Vr] + c"^, or otherwise by a^ + ^r] + y^, 
a'? + ^'v + iK> a"| + /3"77 -f- 7"f, where 



10 


/S = 


10 


7 = 


1 


h' c' 




a' c' 




a' b' 


h" c" 




a" c" 




a" b" 


&c. 




&c. 




&c.* 



On the former supposition the related classes x, y, z, ^, rj, f will be said to 
be cogredient, and on the latter supposition contragredient f. If now we 
have one or more functions of classes of variables so related |, such function 
or system of functions may have associated with it a concomitant, also made 
up of distinct but related classes of variables, such classes being capable 
of being either greater or fewer in number than the classes of the given 
function or system of functions. 

In the primitive function or system, as also in the concomitant, the 
related classes may be all of the same species, or some of one and the others 
of the contrary species. Even if we limit ourselves to the conception of a 

* See my paper in the previous number of this Journal [p. 199 above.] 

t The germ of the notion of contragredience will be found in the immortal Arithmetic of the 

great and venerable Gauss. 

J The relation here spoken of will be observed to be of a dynamical character, not referring 

to the systems as they are in themselves, but to the movements to which they are simultaneously 

subject. 



286 On the Principles of the Calculus of Forms. [42 

primitive function or system of functions with only one class of variables, its 
concomitant may be composed of various classes of variables, in respect to 
some of which it will be covariant with, and in respect to the others contra- 
variant to, the primitive function or system*. This is an immense and most 
important extension of the conception of a concomitant given in my preceding 
paper in this Journal, and will be shown to have the effect of reducing the 
whole existing theory under subjection to certain simple abstract and 
universal laws of operation. 

The relation of concomitance is purely of form. A being a given form, 
B is its concomitant, when A' being derived from A by simultaneous substi- 
tutions impressed upon the class of variables or upon each of the classes 
(if there be more than one) in A, and B' from B by corresponding (coincident 
or contrary) substitutions impressed upon the class or classes of variables in 
B, B' is capable of being derived from A' after the same law as B from A ; 
or, as it may be otherwise expressed, " functions are concomitant when their 
correlated linear derivatives are homogeneous in point of form '\:" 

This definition implies that one at least of the forms must be the most 
general possible of its kind : in a secondary but very important sense, however, 
functions obtained by impressing particular values or relations upon the 
quantities entering into the primitive and its associate form, will still be 
called concomitant. Thus a? — y^ will be termed a concomitant to a^ + y^, 
not that we can affirm that {ax + hyj — (ex + dyj : 

that is (a' -&)a? + Z {afh - c'd) a?y -I- 3 (a&= - cd?) xy^ + {¥ - d') y\ 

treated as a function of x and y, can be derived from {ax -f- hyy + {ex -f- dyf, 

that is (a= + c^) or' + 3 {o?h -1- (?d) x-y + 3 {ah- + cd?) xy- -\- {}f + d^) y\ 

when ad—hc—\ by the same law as {af — y^) from {x^ + y''), fox the elements 
for forming such comparison are wanting, but because a? + y^ and a? — y'^ are 
the correspondent particular values -respectively assumed by 

aa? + 2bx^y + Scxy^ + dy^, 
and its concomitant 

{ad' + 2(f - Ucd) a? - {&l?d - 3c=6 - Zacd) x'y 

-f- {Qac- - 3c6- - Sc&o.) xy- - {a-d -|- W - Sbca) y\ 
when a = l, b = 0, c = 0, d = l. 

With the aid of this extended signification of the term concomitant (whether 
it be a covariant or contravariant) Ave can in all cases speak (as otherwise we 
in general could not) of the concomitant of a concomitant. The relation 

* And of course the concomitant may be an invariant to its origiuant in respect of one or 
more systems of variables entering into the former. 

+ Or, more generally, it may be said that concomitance consists in the persistence of morpho- 
logical affinity. 



42] On the Princijyles of the Calculus of Forms. 287 

between systems of variables has been stated to be Simple ('whether they be 
cogredient or contragredient) when each variable in one system corresponds 
with some one in each other. Compound relation arises as follows : — Suppose 
*> yj ?' V two independent systems of two variables each, and that the 
system of four variables u, v, w, t is subject to linear variations imitating, 
in the way of cogredience or contragredience, those to which x^, xrj, y^, yq 
are subject; then it,, v, w, t may be said to be cogredient or contragredient 
to the continued systems x, y; f , tj. li os, y ; ^, rj be themselves cogredient, 
then a system of only three variables u, v, w, may be cogredient or contra- 
gredient in respect to x^, xrj + y^, yi), and '\i x,y; ^,7] be coincident, u, v, w 
may be similarly related to a^, xy, y". The illustration may easily be 
generalized, and it will be seen in the sequel that its conception of compound- 
relation between systems of a differing number of variables will greatly 
extend the power and application of the methods about to be developed. 
Without having recourse to a fornnal definition, it is obvious that the notion 
of a concomitant conveyed in my former paper in this Journal lends itself 
without difficulty to the most general supposition which can be made of 
functions between which any number of systems of related variables are 
distributed, whatever such relation be, whether simple or compound, and 
whether of cogredience or of contragredience. The proposition stated in my 
last paper relative to a concomitant of the concomitant of a function being 
a concomitant of the original still applies to concomitants in the wider sense 
in which we now understand that term, and the species of each system of 
variables in the second concomitant with respect to the species or either 
species (if there be systems of both kinds in the primitive) will be determined 
upon the general principle which determines the effect of concurrence and 
contrariety being made to operate each upon itself or one in either order 
upon the other. 

The highest law and the most powerful in its applications which I have 
yet discovered in the theory of concomitants may be expressed by affirming 
that when several related classes of variables are present in any concomitant, 
a new concomitant, derived from the former by treating one or any number of 
these classes as independent of the remaining classes, v/i\l still be a concomitant 
of the primitive. I shall quote this hereafter as the Law of Succession. 
This law, to which I have been led up inductively, requires an extended 
examination and a rigorous proof. It is the keystone of the subject, and any 
one who should suppose that it is a self-evident proposition (as from the 
simplicity of the enunciation it might be supposed to be) will commit no 
slight error. 

If (j>{x, y ... z) be any homogeneous form of function of x, y, ... z, every 
homogeneous sum in the expansion by Taylor's theorem of 

(p{u + u', v + v' ... w + w'), 



288 On the Principles of the Calculus of Forms. [42 



which in fact, on making u' = x, v' = y ... w' = z, becomes identical (to a 

numerical factor pres) with {u-j- + v-T-+w-y-\ (f>, is what I have elsewhere 

termed an Emanant, and by a partial method I had demonstrated that every 
invariant of such an emanant in respect to u, v . . . w, in which x,y ... z are 
treated as constants, or vice versd, would give a covariant of (p. The reason 
of this is now apparent. For it may easily be shown* that every emanant 
is in fact itself a covariant of the function to which it belongs with respect 
to each of the related classes of variables which enter into it, or is as it may 
be termed a double covariant. The law of Succession shows therefore that 
a concomitant to an emanant from which one of the classes has disappeared 
will be a covariant of the primitive in respect to the remaining class. 

In applying the law of Succession, great use can be made of a function 
of two classes of letters which may be termed a Universal Mixed Concomitant; 
this is x^ + y7} + . . . + z^, which has the property of remaining unaltered when 
any linear substitution (for which the modulus is unity) is impressed upon 
X, y ... z, and the contrary one upon ^, ij ... ^f. 

If /(«, y) be any function of x, y, of the degree m, f+X (x^ + yrj)"^ will 

* To demonstrate this it is only necessary to observe that if u,v,... w, u', v', ... w' be 
cogredient with themselves and with x, y, ... z, 

<p{u + \u', v + Xv', ... iv + \io') 

will evidently be a concomitant of (j>(x,y, ... z); and, X being arbitrary, the coefficients of the 
different powers of X must be separately concomitants of (p[x,y, ... z), but these coefficients are 
the emanants of 0. Q. e. d. 
t Thus, if 

x = ax' + l>y' + cz', i={gn-hm)^' +{hl-fn)r]' +{fm-gl)!^', 
y=fx' + gy' + hz', r]=(-nb + mc) ^' + ( -lc + na)7i' + {-ma + lb) I;', 
z=zlx' + my' +nz', ^={bh-cg)^' +(cf-ah)i +{ag-bf)t', 

xS + yv + z^=[ f g h \x{x'^' + y'Ti' + z'^') 



a c \ 

f 9 '» )> 
I m nj 



= x'^' + y'Tj' + z'^'. 

When the coefficients of transformation correspond to the direction-cosines between one system 
of rectangular axes and another, the reciprocal system is identical with the direct system ; so 
that X, y, z ; f , ti, f, on this particular supposition, may be regarded indifferently as contragredient 
or as cogredient; accordinglythey may be made identical, and then x^ + y'^ + z'^ remains invariable, 
which is the well-known characteristic of orthogonal transformation. It may be observed here 
that there exists a special theory of concomitance limited to such species of linear transform- 
ations, which may be termed Conditional Concomitance, and I have found in several cases that 
the invariants of conditional concomitants turn out to be absolute invariants of the primitive. 
Much more important is the remark that there exists a theory of universal concomitants for 
an indefinite number instead of merely two systems of variables, as used in the text. In 
the sequel it will be seen that the application of this universal concomitant (like the touch of 
an enchanter's wand) serves to transmute covariants into contravariants, and back again, and 
causes single invariants to germinate and fructify into complete connected systems of forms. 



42] On the Pi-incvples of the Calculus of Forms. 289 

be a mixed concomitant of /, it being evident that every function of con- 
comitants of a function is itself a concomitant of the same. 

Suppose now 

y= ax™ + mhx''^~^y + \m (ja — l) cx^~'^'if + &c., 
the concomitant becomes 

(a + Xf '") a;™ + m (6 + X^'^~\) x'^~^y + \m (m — 1) (c + \%"^~'^"rf) + &c. 

Consequently if P be any concomitant of /, P' obtained from P by writing 
a + X^™, 6 + \^'"-~^rj, &c. for a, 6, &c., will still be a concomitant of f; and by 
Taylor's theorem P' evidently equals 

+ &C. 

If we take P an invariant of /, we have M. Hermite's theorem* for 
f(x, y), and precisely the same demonstration applies to the general case 
of y (a;, y ... z). P' is, by virtue of the general rule, a contravariant of / in 
respect to f, 77 ... ^: if P be taken a function containing one single system, 
and is also a contravariant to / in respect to that system, P' will be a double 
contravariant ; and if we make the two systems in P' identical, we have the 
extension of M. Hermite's theorem alluded to by me in one of the notes f 
to my last paper, wherein I have stated that " / may be taken any covariant 
of the function" : as regards the purpose of that statement, the word covariant 
WAS used in error for contravariant. 

The preceding method may be viewed as a particular application of the 
general principle, that if Ui, U^... Um be any m functions (whether con- 
comitants any of them of the others or not), then any concomitant of 
X-JJ-i^ + XjJ^ + ... + \m.Um being expressed as a function of Xi, X2 ... Xm, 
every coefficient in such expression will be a concomitant of the system 
?7i, f/j ... Um- Thus, for example, if U and V be two quadratic functions 
of 11 variables x, y ... z, the discriminant a(\U + [iV) will contain n+\ terms, 
of which the coefficients of the first and last will be D C/" and D V; and every 
one of the (» + 1) coefficients will be a concomitant (of course an invariant) 
of U and V. These (n -\- 1) invariants will in fact constitute the fundamental 
scale of invariants to the system U and V, and every other invariant of U 

* This theorem was first stated to me by Mr Cayley, who, I understand, derived it from 
M. Eisenstein, under the form of a theorem of covariants, which of course it becomes on inter- 
changing X, y with - y, x. But as a theorem of covariants it could not be extended to functions 
of more than two variables. M. Hermite appears to have discovered this theorem, under its 
more eligible form, subsequently to, but independently of, M. Eisenstein. 

[t p. 201 above, note *.] 
s. 19 



290 



On the Principles of the Calculus of Forms. [42 



and V will be an explicit rational function of the (m + 1) terms of the scale. 
In connexion with this principle may be stated another relative to any system 
of homogeneous functions of a greater number of variables of the same class, 
namely, that if any set of the variables one less in number than the number 
of the functions be selected at will, and any invariant of a given kind be 
taken of the resultant of the functions in respect to the variables selected, 
all such invariants so formed will have an integral factor in common, and 
this common factor will be an invariant of the given system of functions. 

It will be convenient to speak hereafter of systems for which the march 
of the linear substitutions is coincident as cogredient, and those for which 
the march is contrary as contragredient systems. 

Suppose m cogredient classes of m variables, the determinant formed by 
writing the m x in quantities in square order will evidently be a universal 
covariant. Thus, take the two systems x, y; ^, ?;. xt] —y^ is a universal 
covariant, and evidently therefore F, which I use to denote 
(j) (x, y)X(f) (^, 7?) + X (xri - y^)'^, 

will be a covariant to ^ (x, y). Let ^ {x, y) be of m dimensions ; any invariant 
of i'^ will be an invariant of <^; thus, let the two systems x,y\ ^, 17 be treated 
as perfectly independent, and take the discriminant of i'' (viewed as a function 

of x,y; ^, rj), that is the resultant of the four functions -5- , -f , ~Ty , -f > 

this resultant will be an invariant of <^; and X being arbitrary, all the 
coefficients of its different powers will be invariants of (p. We thus fall upon 



another theorem of M. Hermite, namely that if X = 



(j) (x, y)x4> (^, v) 



the 



{x^-yvT 

coefficients of the equation which will give the minimum values of X are 
invariants of cj). So more generally, any invariant of /(.«, y, f , 7?) — X (x^ — yrj)^, 
f being of the degree m in x, y and in ^, 77, will be an invariant of /; and 
among other invariants may be taken the discriminant obtained by treating 
X, ^, y, 7] as absolutely unrelated. 

If / be a function of various classes each containing w covariables, and if 
not less than n of these classes be covariable classes, and after selecting at 

will any n of such systems, as x^, y^ ... z^; x„,y^...Zo\ Xn, y-n.--- ^n, the 

symbolical determinant 



d 


d 


d 


dxi ' 


dyi ' 


' dzi 


d 


d 


d 


dx^' 


dy,' 


" dz^ 




d 


d 


d 


dXn' 


dyn' 


' dzn 



42] On the Principles of the Calculus of Forms. 291 

be expanded and written equal to D, then D'/ will be a concomitant of/; and, 
more generally, by selecting different combinations of the covariable systems 
n and n together in every way possible, and forming corresponding symbols 
of operation E, F...H, we shall have B' .E '■'... H^'-Kf, for all values of 
t, t' . . . (i), a covariant of/" in respect to the classes so combined. This explains 
and contains the whole pith and marrow of Mr Cayley's simple but admirable 
method of obtaining covariants and invariants (or, as termed by their author, 
hyperdeterminants) to a function <^i of a single system oci, y^.-.z^; he forms 
similar functions cj)^... 4>^ oi x^, y, ■■• ^2] •■■ x^, y^ ■■■ z,,., and uses the product 
</>! X (^2 X • ■ • X (/)^ as a function/ of ^ systems : the multiple covariant obtained 
by operating thereupon becomes a simple covariant on identifying the 
different classes of covariables introduced in the procedure. 

Section II. On Complex Concomitance. 

We have hitherto been engaged in considering only a particular case 
of concomitance, the true idea of which relates not to an individual associated 
form (as such), but to a complex of forms capable of degenerating into an 
individual form. Such a complex may be called a Plexus. A plexus of forms 
is concomitant to a given form or combination of forms under the following 
circumstances. 

If (0) be the originant, meaning thereby the primitive form or system of 
forms, and P the concomitant plexus made up of the fj. forms Pj, P2 ... P,x, 
and if, when by duly related linear substitutions, becomes 0', the plexus 
P becomes P', made up of the forms P\, P\ ...P'^, and if the plexus 'P 
formed from 0' after the same law as P from be made up of the forms 
'Pj, 'P2 ... 'P(i, then will each form in either of the plexuses 'P, P' be a linear 
function of all the forms in the other plexus, and the connecting constants in 
every such linear function will be functions of the coefScients of the substitu- 
tion whereby and P have become transformed into 0' and P'. 

A function forming part of a concomitant plexus may be termed a 
concomitantive. Concomitantives therefore usually have a joint relation 
to a common plexus and a concomitant is only another name for an unique 
concomitantive. Every plexus contains a definite number of concomitantives; 
in place of any one of these may be substituted an arbitrary linear function 
of all the rest, but the total number of independent forms sufficient and 
necessary to make the complete plexus respond to the requirements of the 
definition will remain constant. 

If now we combine together the whole number of functions contained 
in one or more plexuses concomitant to any given originant, all of the same 
degree relative to any given selected system or systems of variables, and if 
the number of the concomitantives so combined be exactly equal to the 

19—2 



292 On the Principles of the Calculus of Forms. [42 



number of terms in each, arranged as a function of the selected class or classes 
of variables, then the dialytic resultant (obtained by treating each combination 
of the selected variables as an independent variable, and forming a deter- 
minant in the usual manner), will be a concomitant to the given originant. 
This, which is only the partial expansion of some much higher law, may 
be termed the " Law of Synthesis." 

Let y be any function of a single class of variables a^i, «;„...«„. Let y^ 
represent any product of these variables or of their several powers of any 
given degree r ; the number of different values of ■)(^ will be /i, where 
n (w + 1) ...{n-\-i — 1) 
^" 1.2...r • 

and 'X^if, 'x^if ... %^/will form a covariantive plexus to/. 

Again, let '^ represent any product of the degree r of the symbols 
d d d 

dxi ' dx^ ' ' ' dxn ' 
&i/, ^f ... ^^/will also form a co variant plexus to f. 

The coefficients of connexion between the forms of either plexus depend 
in an analogous manner upon the coefficients of the substitution supposed 
to be impressed upon the variables, with the sole difference that every 
coefficient taken from the line r and column s of the determinant of sub- 
stitution which appears in any coefficient of connexion of the one plexus 
is replaced by the coefficient taken from the line s and the column r in the 
corresponding coefficient of connexion for the other plexus. 

Let/(a;, y) be any function of x, y of the degree 2m ; then 

/dy" (dV^'^ d_ fdY 

\dxj ' \dx) dy' \dy) 

will form a covariantive plexus ; thus, suppose 

f{x, y) = a^x""^ + 29;ia,«™-i y+... + a„,n+,y°-'^ ; 
omitting numerical factors, the plexus will be composed of the (m + 1) lines 
following : 

Ois;'" +ma.,x™-^y +...+am+jy™, 

a^x™- + ma^x^'^y -\- ... + am+iy™< 



«,„,+!«'" -I- ma^+._x^-^y -I- ... -I- aam+iy™, 
and consequently, by the law of synthesis, the determinant 

ai , Oo CLm+ 

(^2 J ^3 ^m-(- 



is an invariant off. 



^m+i ) <^wt+2 ■ • • ^2m+i 



42] On the Principles of the Calculus of Forms. 293 

When this determinant is zero, I have proved in my paper* on Canonical 
Forms, in the Philosophical Magazine for November last, that / is resoluble 
into the sum of m powers of linear functions of x and y. I shall here- 
after refer to a determinant formed in this manner from the coefficients 
of / as its catalecticant. Mr Cayley was, I believe, the first to observe that 
all catalecticants-f- are invariants. 

Again, more generally, \ebf(so, y, ^, t?) be a function of the mth degree 
of as, y, and of a like degree in respect of ^, t], which are supposed to be 
cogredient with x and y ; then 

/(*. y> ^> '?) + A. {xv - y^T 

(say F) will be a concomitant of/; and therefore if we take the system 

\dx) ' \dxj dy \dy/ ' 

which will be functions of f and t] alone, and take their resultant, this 

resultant will be an invariant of/. As a particular ease of this theorem, let 

. f^d cZ\™^ 

■^=[^dx + '>dy) '^' 

where ^ is supposed to be a function of x and y only and of 2m dimensions, 
/ is a concomitant of j), and therefore the invariant of/ obtained in the 
manner just explained, will be an invariant of (p. Thus then we have an 
instantaneous demonstration of the theorem given J by me in the paper of 
the Philosophical Magazine before named, namely, if 

{x, y) = ai««» + ^ma^oF^^y + . . . + a.^+^.y^"', 

say, in order to fix the ideas, = aoifi ■{■ Qhafy + locx'^y"- + . . . + gy^ ; then the 
determinant 

a, b, c, d + \ 

b, c, d — ^\, e 

c, d + ^\, e, f ■ 
l-\ e, / g 

(and the analogously formed determinant for the general case) will be an 
invariant of (p. The general determinant so formed is peculiarly interesting, 
because it furnishes when equated to zero the one sole equation necessary 
to be solved in order to be able to effect the reduction of (f> (x, y) to its 
canonical form, and gives the means, irrespective of any other view of the 
theory of invariants, of determining completely and absolutely the condition 

[* See p. 282 above.] 

+ But the catalecticant of the biquadratic function of x, y was first brought into notice 
as an invariant by Mr Boole ; and the discriminant of the quadratic function of x, y is identical 
with its catalecticant, as also with its Hessian. Meicatalecticizant would more completely express 
the meaning of that which, for the sake of brevity, I denominate the catalecticant. 

[J p. 277 above.] 



294 On the Principles of the Calculus of Forms. [42 

of the possibility of two given functions of the same degree of x, y being 
linearly transformable one into the other. This theorem will be obtained 
in a more general manner in the following section. I only pause now to 
make the very important observation, that not only is the determinant an 
invariant, but every minor system* of determinants that can be formed from 
it (there are of course m such systems) is an invariantive plexus to the given 
function ^. 

The form under which this theorem presents itself suggests a theorem 
vastly more general and of peculiar interest, as showing a connexion between 
the theory of functions of a certain degree and of a certain number of 
variables with other functions of a lower degree but of a greater number 
of variables. Here again, under a different aspect, is reproduced the great 
principle of dialysis, which, originally discovered in the theory of elimination, 
in one shape or another pervades the whole theory of concomitance and 
invariants. 

Let j> represent any function of the degree pq (of any number, or, to 
fix the ideas, say of three variables x, y, z) ; let the general term of </> be 
represented by 

(i.2...«)(i^^2"..^.W(?-2...7) («' ^' ^) "^y'''' 

where a + /S+y^pq, and (a, /3, 7) represents a portion of the coefficient of 
x'^y^zy. 

Let 

1.2...P r^t-f) 

(1.2...r)(1.2...s)(1.2...0*^^~ '■■'■" 
where r + s + t=p, so that there are as many ^'s as there are modes of 

* These minor systems mean as follows : — the system of rth minors comprises all the distinct 
determinants that can be got by striking out from the square array (which I call the Matrix) 
from which the complete determinant is formed, any r lines and any r columns selected at will. 
The last, or «ith minor, is of course a system consisting of the coefficients of {x, y), and it is 
evident that if <fi(x,y ... z) be any function of any number of variables x, y ... z, the coefficients 
will form an invariantive plexus to (f>. 

The following remark as to the changes undergone by the coefficients of when the variables 
undergo any substitution, is not without interest and importance for the theory. 

Let X become fx +f'y + ... + (f) z, 
y 9x + g'y+... + (g)z, 

z hx + li'y+ ... + (h)z. 

Then the coefficient of the highest power of x becomes 

<t>{f,g.:h), 
and the coefficient of the term containing y' ...z^ becomes 



42] On the Principles of the Calculus of Forms. 295 

subdividing p into three integral parts (zeros being admissible); that is 
^ (jP + 1) (p + 2) (p + 3). Then any product such as x'^y^z* may be divided 
in a variety of ways into the product of q of these ^'s, and it may be shown 
that the entire quantity 

pq ipq -l).-.l 



(1.2...a)(1.2...;8)(1.2...7) 
1.2. 



{oc^y^zy) 



= 1 



K'"^)], 



l(1.2...mi)(1.2...m0...(1.2...m^)^ "' ^ 

where mj 4- ma + . . . + m,. = q. Consequently may be represented under the 
form of a function of the degree q oi ^{p + l)(p + 2) {p + 3) (say i) variables 
6i, ^2 ... 0,,, and its general term will be of the form 

(1.2...mO(l.V.^m!)...(1.2...m.) ^«' ^' '>'> i^"'^" - ^"^i' 
where a, /3, y are the indices respectively of x, y, z, when the last factor 
is expressed as a function of these variables*. Now if ^ be used to denote 
this new representation of </> when ^i, d^--. 6, are treated as absolutely 
independent variables, and if we attach to it any universal concomitant, as 
{x^ +yrj+ z^y admitting of being written under the form w (^i , O^... 6,), 
wherein the coefficients will be functions of ^,17, i^; then any invariant to 
& and &), treated as two systems of i variables, will be a concomitant to ^, 
the original function in x, y, zf. ^ and (o may be termed respectively, for 
facility of reference, the Particular and Absolute functions. Thus, for example, 
we take ^ a function of x, y of the degree 4re, say 

a^x^"' + ina.y-'' y + &c. + aj^+i^/^", 
and make p = 2w, g = 2, so that ^ becomes a quadratic function of (2?i + 1) 
variables obtained by making «^" = ^i, x™-'^y = d^ . . . y"-'^ = O^n+ity and the 
concomitant &>, formed from {^x + rjy)^, becomes 

6I1P' + 2ne,^''~'v + •■■+ O.n+iV'"' ; 
then if we take R the quadratic invariant of to, that is 

1 2 T (^n) 
E = e,0^+, - 2n0,d^ &c. ± W^' ny * ^^"+'^'' 

* See Note (1) in Appendix, [p. 322 below.] 

t In fact a- is a concomitant to 0, and w to a power of the universal concomitant ; the 8's 
forming a system of fariables cogredient with the compound system x''iy',z'i, x'^2y'^2z'i, (fee: 
and it must be well observed that the same substitutions which render S- and u respectively 
identical with <p and a power of the universal concomitant, would render an infinite number 
of other functions also coincident with the same ; but none of these other functions would be 
concomitants. Herein we see the importance of the definition and conception of compound 
relation ; the 6 system being compound by relation with the x, y, z system, after the manner 
of cogredience. 

J A slight variation upon the method as above explained for the general case has been here 
introduced inadvertently by writing x^''-'j/ = 9„, &c., in lieu of 1m^^~^y = 6^, <fec., which, as it 
does not in any degree affect the reasoning, I have not deemed it worth while to alter. 



296 On the Principles of the Calculus of Forms. [42 

it will readily be seen that the determinant of S- + XR, treated as a quadratic 

function of (2n + 1) variables, will give an invariant of ^, and this will be the 

same as that obtained by the particular method above given. Thus, suppose 

(«, y) = aac^ + ^hai?y + Qcx^y'^ + 4<dxy^ + ey*. 

Let x' = Oi, 2xy = 6^, y- = 6^, 

^ = aO:,^ + 2be,e, + cdi + "led A + "^dOA + e(9/, 
a = {x^ + yrff = x'd-i + xyOo, + y-O^, 

R = 6i6s —-T- 

Then A the discriminant of ^ + 2\R in respect to 6^, 6^, 6, 
a, h, c + X 

h, c— |-X, d 
c+X, d, e 

and I may remark that the relations between the several transformees of the 
invariantive plexuses formed by the minor determinant systems of A (in this, 
and in general for the case of an evenly-even index) may be found by treating 
^ + 2XR as a quadratic function of the variables (in this case Oi, 6^, 6^) and 
applying the rule given by me in the Philosophical Magazine in my* paper 
" On the relation between the Minor Determinants of linearly-equivalent 
Quadratic Forms."f This second method, however, is not immediately 
applicable to the case of indices oddly even, that is of the form 4>n + 2, to 
which the first method applies, equally as to the case in ; for if we make 
2n + l =p and q = 2, ta being of an odd degree, has no quadratic invariant ; 
it has however a quadratic covariant, which will be of the second degree 
in respect to ^j, 02 ■■■ 0p+i ^■s well as in respect to ^, T, and if we call this R 
and take the discriminant of ^ -i- XR in respect to the variables Oj, 6^... 6p^j , 
we shall obtain, as I am indebted to a remark of my valued friend M. Hermite 
for bringing under my notice, a very beautiful and interesting function of X, 
of which all the coefficients will be contravariants of <(>. Thus, let 
<f) = ax^ -\- %hoi?y -l- Yoca^y"^ -f- 20da?y^ + loex^y* + Qfxy'^ + gy^, 

[* p. 241 above.] 

t Moreover, upon the supposition made in the text, the particular and absolute functions 
a- and w may be treated in all respects as if they were functions characterizing quadratic loci, 
and any singularity in their relation will correspond to and denote a singularity in the given 
function to which 3- refers. Thus, for instance, if be a function of x, y of the eighth degree, 
3- and u will be quadratic functions of five letters each. Quadratic loci have no other singularity 
of relation than what corresponds to different species of contact. The number of contacts between 
loci, characterized by 5 letters, is 24 (see my paper J in the Philosophical Magazine, " On the con- 
tacts of lines and surfaces of the second order"). Consequently this mode of representing 3- and a 
will give rise to the discovery and specification of 24 different kinds of singularity in 0, and the 
analytical characteristics of each of them. But there of course may, and in fact will, exist 
other singularities in besides those which have their correspondencies in the relations of these 
quadratic concomitants. [J p. 237 above.] 



42] 



On the Principles of the Calculus of Forms. 297 



make a? = e^, 2x^7/ = 0^, ^xif = 6^, y^^d^, ' 

so that 

S- = a^/ + 2bdA + cOi + 2c6A + ^dOA + ^dO^O^ + gd," + 2/0 A + e0i + 160^0^, 
a={x^+ yiif = 6lip + 6l,f r; + 0^^7f- + 0^if, 
R= 30,^ + 0,7,, e,^+ 0,7j 
02^ + 0sV. 0,^ + S0,V 

-R = ^A^ + iA- + ^vdA - Q^v0A - HA03 - H^A- 

Consequently the discriminant in respect to 0^, 0o, 0s, ^4 of S — 2\R becomes 



a, 


h, 


c - 3\p, d - 9X|^7? 


b, 


c + 2\^\ 


d + X^rj, e — ZX'rf 


c - 3Xp, 


d + \^v. 


e + 2\77^ / 


d - 9X^71, 


e - 3\n\ 


/. 9 



If this determinant be expanded as a function of X, all the coefficients of the 
various powers of X will be contravariants to-the given function <^. The term 
involving X* is zero. Let ^ become — y and t] become x, then the remaining 
terms (abstraction made of the powers of X) become covariants of <^. The 
first term (the coefficient of X^) becomes <^ itself; the last term is the 
catalecticant, and thus we see, in general, that for functions of x and y 
of an oddly-even degree, a whole series of covariants may be interpolated 
between the function and its catalecticant, the dimensions in respect of the 
coefficients of in arriving at each step increasing by 1 unit and the degree 
in respect of the variables diminishing by 2 units. This is consequently 
a much simpler and more available scale than one with which I have been 
long previously acquainted, and which applies alike to functions of any even 
degree. 

Thus, let (^ {x, y) be of 2h dimensions ; form all the even emanants of ^ 

/ d d\^'- 

which will be all of the form [^ -j — ^Vrf) 4'> ^^^ take their respective cata- 

lecticants in respect to ^ and 97. We shall in this way obtain a regular scale 
of covariants interpolated between the Hessian of <f) (corresponding to t = 1) 
and the catalecticant of cf> (corresponding to t = k). If be of the degree 
2k + 1, we shall have an analogous scale interpolated between the Hessian 
of (p and its canonizant ; the latter term denoting the function which is the 
product of the k + 1 linear functions of x and y, the sum of whose (2k + l)th 
powers is identically equal to cj)*. 

By means of the Theory of the Plexus we may obtain various representa- 



* See Note (2) in Appendix, [p. 322 below.] 



298 On the Principles of the Calculus of Forms [42 

tions of the same invariant; thus, for example, if we take F a function 
of X, y of the fifth degree and form its Hessian H, that is 

tPF d'F 

da? ' dxdy 

d-'F d?F 
dydx ' dy'' 

this will be a function of the sixth degree in x, y, and of the two orders 
in the coefficients. If we combine the two plexuses 
dF dF d^ dm^ d?H 
dx' dy' dx^ ' dxdy ' dy" ' 
we shall have five equations between which ar*, a^y, oo'y'^, x-tf, y^ may be 
eliminated dialytically ; the resultant will be of the 2 + 3.2, that is the 
eighth order in the coefficients, and of the form qF — I I, where n F and I^ 
are respectively the determinant and quintic invariant of F, each affected 
with a proper numerical multiplier (the "5 — J.=" of my supplemental* essay 
on canonical forms) which, as Mr Cayley has remarked, may also be repre- 
sented by the resultant of P ; — ; ~ where P and Q are respectively the 
dx dy 

d d 



quadratic and cubic invariants in respect to f and 'J of ( ^ j" + '7 j~ ) ■^• 

It will be well at this point to recapitulate in brief a method of elimination 
applicable to certain systems of functions published by me many years since 
in the Philosophical Magazine, and to compare this method with that afforded 
by the theory of the plexus for finding an invariant for each of the very same 
systems, possessing all the external characters, formed in a precisely similar 
manner to, and not impossibly identical with, the resultant of every such 
system. I shall devote my first moments of leisure to the ascertainment 
of this last most important point, as to the identity or otherwise of the 
plexus-invariant with the resultant. Take the case of three functions of 
X, y, z (say <^, yjr, m) each of the same degree n; to fix the ideas, suppose 
71 = 3: there are two purely algebraical processes (modifications of the same 
method and leading to identical results) by which the resultant of (f>, 1/^, a> 
may be found. I shall call these processes the first and second respectively. 

First process : Write 

(f, = x'P +yQ +zR, 

^}r = a^P' + yQ' + zE, 

CO = x-P" -f yQ" + zR" , 

decompositions which may be effected in an infinite variety of manners, 
so that P, Q, R shall be integer functions of x, y, z; take the linear resultant 
of ^, i|r, ft), in respect to x-, y, z, which call H^^ 1^ ,; this will evidently be 

[* p. 205 above.] 



42] On the Prmciples of the Calculus of Forms. 299 

of 9 — 4, that is, of 5 dimensions. Form analogously the functions -Efi,2,i, -^1,1,2; 
jETa,!,!, .ffi,2,ij -^^1,1,2 constitute an auxiliary system of functions which vanish 
when (j>, yjr, o) vanish together; combine this auxiliary system with the 
augmentative system 

x^ip, y-(j), z-(f>, ocy<^, yzcj), zx(^, 

x^'d), y^o), z''eo, ocyco, yzw, zxa, 

x^yjr, y^-\lr, z^-^, xyyjr, yzyfr, zxyjr. 

We shall thus have in all 3 + 3 x 6, that is, 21 functions into which the 
21 terms of, x^y, a^z, &c. enter linearly : the linear resultant of these 21 
functions is the resultant of </>, i|r, m, clear of all extraneousness. 

Second process : Write 

j>=a?P +yQ +zR, 
yjf = x^P' + yQ' + zR', 
m =afF" + yQ" +zR", 

and, as before, take the linear resultant .£^3,1,1, which will however be of 9 — 5, 
that is, of only 4 dimensions. 

Again, take 

<f)=x'L + y^M + zN, 

y}r = x^L' +y^M' +zN', 

CO = x-L" + y"-M" + zN", 

and form the determinant ^2,2,1 ; we shall thus have the auxiliary system 

■^3,1,1, -9^1,3,1, ^],1,3J -^2,2,l> -ff2,l,2> ffl,2,i- 

Let this be combined with the augmentative system 

xa>, ya, zoo ; X(^, ycf>, zcj} ; xyjr, y-^jr, z^lr. 

Between these 6 + 9, that is, 15 functions, the 15 terms a^, a^y, a?z, &c. may 
be linearly eliminated, and the resultant thus obtained will be precisely 
the same as that got by the preceding process. 

Here we have 6 auxiliaries and 6 augmentatives ; the auxiliaries are 
of three dimensions in respect to the coefficients of (f>, yfr, co; the augmenta- 
tives of one dimension only ; in the former process there were 3 auxiliaries 
and 18 augmentatives, 6 x 3 + 9 = 27 = 3 x 3 + 18. 

Now let this method be compared with the following : 

First process : Take the 18 augmentatives x-cf), x-00, ay'\]r, &c. as in the first 
process of the algebraical method above explained ; but in place of the 
3 auxiliaries therein given, take another system of 9 as follows: 



300 On the Principles of the Calculus of Forms. 



[42 



Write the determinant 



dx' 


dy' 


dz 


d^lr 

dx ' 


dy ' 


df 
dz 


doj 


dm 


dco 


dx' 


dy' 


dz 



= -R; 



r/7? ^7? 

-1- , "T— form a concomitantive plexus; the 18 augmentatives form 
dy '^'' ^ 



dR 

dx ' dy' dz 

another; the linear resultant of these two plexuses will be an invariant 
of 0, t^, CO, and of precisely the same dimensions as the resultant last found ; 
if they are not identical it will be indeed a matter of exceeding wonder, 
and even more interesting than if they should be proved so to be. 

Second process : Combine the augmentative plexus 

xw, ya>, zu> ; x(f), yep, zcp ; x-^, y\{r, zifr, 

with the differential plexus 

d"-R d^R (PR d^R d'R d-R 

dx" ' dxdy' dy^ ' dydz' , dz- ' dzdx' 

we thus obtain a linear resultant in a manner precisely similar to that 
afforded by the second process of our algebraical method. 

In general, if (j), yjr, a be of the degrees n, n, n, as there are two algebraical 
varieties of the linear method for finding the resultant, so are there two 
varieties of the concomitantive method for finding the resembling invariant. 
In both methods the augmentatives are identical ; the only difference being 
in the auxiliary system. 

In the first process the augmentative system will be got by operating 
upon each of the functions 0, -v/r, m, with the multipliers a;"~^, y'^"'^, z^"^, 
and the other homogeneous products of x, y, z ; the auxiliary system by 



operating upon R with the symbolical multipliers (-3- 



d_y- 



c/ 



d d d 



dy-"- 
dy) 



of the decree n — 2. 



and the other homogeneous products of -, , -j- , t~ 

° '■ ax dy dz 

In the second process the augmentative system is formed by the aid 
of the multipliers a;"""^, 2/"~^ 2"~^, &c., and the auxiliary system by aid of 



dx] 



fdy'-' 
\dy) ■ 



ydzJ 



&c. 



For the particular case of n = 2 the first process of the concomitantive 
method is merely an application under its most symmetrical form of the first 



42] On the Principles of the Calctdus of Forms. 301 

process of the general algebraical method. The second process of the con- 
comitantive method for this same case (at least when ^, i^, co are the partial 
differential coefficients of the same function of the third degree) has been 
shown by Dr Hesse to give the resultant, so that for this case, at all events, 
we know that each concomitantive auxiliary must be a linear function of the 
augmentatives and the algebraical auxiliaries. 

Again, if we go to the system where ^, i/r, a are of the respective degrees 
n, n, n+1. In the algebraical method (for applying which there are no 
longer two, but one only process), the augmentative system is obtained 
by multiplying </> by the homogeneous products of a;™"^ a;"~^y, ai^~^z, &c., 
yfr by the like products, and a by the homogeneous products x'^", x'^'^y, &c. 
The auxiliary system is made up of functions of the general form 

-ffj),5,r where p + g' + r = n+2, 

Jip^q^r being the determinant obtained by writing 

(/. = LxP + Myi + Nz", 

■f = L'xP +M'yi + iVV, 

CO = L"aiP + M"yi + i^V. 

And in like manner for the case of </>, i/r, to, being of the respective degrees 
71, 11, n—1, the augmentative system is obtained by affecting <p, ■yjr each with 
multipliers «"~^, x'^~^y, &c., and w with the multipliers a;"~^, x'^~^y, &c. 

The number of functions (for either case) in the augmentative and 
auxiliary plexuses thus obtained will be found to be exactly equal to the 
number of terms in each such function, as shown by me in the paper alluded 
to. Let this be compared with the transcendental method (I use this word 
at this point in preference to concomitantive, because in fact the algebraical 
and differential auxiliary systems are both alike concomitantive plexuses 
to <})). For the case of n, n, n + 1, the Jacobian determinant R of 0, i^, to 

will be of the degree 3?) — 2, and the system \-r\ R, [-t-\ \-^\ R, &c. 

combined with the augmentative systems 

«"~^a), a;"~^!/a), &c. 

«"~^(^, x^~^ycj), &c. 

will give an invariant resembling (at least in generation and form) if not 
identical with the resultant of (p, ^jr, co. For the case of (f>, i|r, co being of the 
degrees n, n, n — 1, the Jacobian R is of the degree 3?i — 4 and 

(A-Y~"'r (sLY'" a r ^c 

\dx) ' \dxj dy ' 



302 On the Principles of the Calculus of Forms. [42 

is the system which, combined with the augmentative systems 

^n-2^^ X^-^yy^r, &C. 

a;"~^(u, x'^~-ya), &c. 

will produce the resembling invariant. 

Finally, for the last and more special case which the algebraical method 
applies to, namely of (j), ■x/r, w, 6, four quadratic functions of x, y, z, t, there 
can be here little doubt (upon the first impression) that in place of the 
algebraically obtained plexus 

•"2,1,1,1) -"l,!,l,l> -"l, 1,2,1! -"1,1,1,21 

may be substituted the differential plexus 

dR dR dR dR 
da; ' dy ' dz ' dt ' 

which, combined with the augmentatives 

x(f>, x\jr, XQ3, xO; y(f), yyjr, ya, yd; z(p, z-^jr, za, zd; t(f>, t-\lr, to), tO, 

will render possible the dialytic elimination of the 20 homogeneous products 

a?, x"y, x^z, a?t, xyz, y^, &c. &c.* 

Upon precisely the same principles may be verified instantaneously 
the method given by Hesse (without demonstration) for finding the polar 
reciprocal of lines of the third and fourth orders, at least to the extent of 
seeing that the functions obtained by his methods are contravariants (of the 
right degree and order) of the function from which they are derived. The 
polar reciprocal to a surface of the third degree may be obtained in the same 
manner. 

Let (p {x, y, z, t) be the characteristic of such a surface. If we form 
a differential plexus of the first emanant of <p taken together with the 
concomitant w = x^ + yrj + z^ + td, by operating with 

d d d d I i.1 d , d ^., d n, d\ , , ^ . 

dx' dy' dz' dt '''^°'' [^ dx+ "^ Ty"- ^ dz"" ^ dt)^'^ + ^'"^' 

and combining this plexus with x^' + yr] + zt,' + t& , the resultant taken in 
respect to ^', ??', t,' , 6' (say R) will (according to the law of synthesis) be a 

* Subsequent reflection induces me to reject as very improbable the (at first view likely) 
conjecture of the identity of the resultant with the invariant which simulates its form, except in 
the proved cases of three quadratic functions and the strongly resembling case of four quadratic 
functions last adverted to in the test above. Did this identity obtain, analogy would indicate 
that the catalecticant of the Hessian of two homogeneous functions of the same degree in x, y 
should be identical with their resultant, which is easily demonstrated to be false, except when 
the functions are of the third degree. 



42] On the Principles of the Calculus of Forms. 303 

contravariant to the system (/> + \w and w, and therefore to 0, because w is 
itself a concomitant to <^. R is of the third degi-ee in x, y, z, t, as also in the 
coefficients of 0. If we form a differential plexus of R + fiw analogous 
to that formed above with <j> + Xw, and combine these two plexuses with the 
augmentative system xw, yw, zw, tw, there will be 4+4 + 4, that is, 1 2 
functions containing the 12 terms x", y'^, z-, t^, xy, xz, xt, yz, yt, zt, \ fx,, and 
the dialytic resultant, which will be found to be a contravariant of the twelfth 
degree in f , r], ^, 0, and of the twelfth order in respect of the coefficients of <f), 
will be (there can be little doubt) the polar reciprocal to the characteristic <f>. 

A few remarks upon the analytical character of a polar reciprocal may be 

not out of place here. If ^ be any homogeneous function of the degree m 

of any number (n) of variables (x, y ... z), the object of the theory of polar 

reciprocals is to discover what is the relation between ^,r} ... ^ expressed in the 

simplest terms such that, when this equation is satisfied, ^x + rjy + ... + ^z = 

will be tangential to <^ = 0. In order for this to take effect it is necessary 

that when any one of the variables z is expressed in terms of the others ... y,x, 

and this value established in cf>, the discriminant of ^, so transformed, should 

be zero. Consequently the characteristic of the polar reciprocal to (p is 

that rational integral function which is common to all the discriminants 

obtained by expressing ^ (by aid of the equation ^x + 'rjy+ ... +^z) as a 

function of any (n — l) of the variables. Let I^ be any invariant whatever of 

the order r of (f>x (meaning by this last symbol what <f) becomes when x is 

eliminated), and ly ... I^ the corresponding invariants when y ... z respectively 

E„ 
are eliminated ; I^ will evidently be of the form ,,. ^^ , the numerator being 

an integer of r dimensions in the coefficients of ^ and of 7nr dimensions in 
respect of ^, r} ... f ; and by the fundamental definition of invariants it may 
easily be shown that 

11 1 ^ 



and therefore 



^1 ^)-Pi ^^ 



Ex Ey E^ , m (n -2)r 



Consequently all these quotients must be essentially integer, and any one 
of them will be of the order r in respect of the coefficients of (p and of the 

* We see indirectly from this, tliat for a function of (™ - 1) , say 7, variables of the degree m, 

an invariant of the order r must be subject to the condition that — = an integer. This is easily 

7 

shown upon independent grounds ; when 7 = 2, — must be not merely an integer but an even 

7 
integer, and doubtless some analogous law applies to the general case. 



304 071 the Principles of the Calculus of Forms. [42 



mr 



degree = in respect of f, 77 ... f. Consequently the polar characteristic 

of ^, which is the common factor of the discriminants of Zj, ly ... I^ (for which 
species of invariant r evidently is equal to (m — 1) {m — 1)""^, the function being 
in fact the discriminant of a function of the mth degree of (n — 1) variables), 
will be of the order (?i — l)(m— 1)"~^ in respect of the coefficients of 
and of the degree m (ni — 1)"~- in respect of the contragredients ^, t] ... ^. 

As to what relates to the reciprocity which exists between tJ3 and its polar 
reciprocal ^jr, this is included in a much higher theory of elimination, one 
proposition of which may be enunciated somewhat to the effect following, 
namely that if ^ be a homogeneous function of x,y ... z, and oj of x, y ... z, 
u, V ...w, and if, by aid of the equations 

^ = 0, 

#+X — = 
dx dx ' 

dy dy 



dz dz 



x,y ... z be eliminated and the resultant be called i|r, then the effect of 
performing a similar operation upon i/r, oi, with respect to m, w ... w, as that 
just above indicated for the system ^, co, with respect \.o x,y ... z, will be to 
give a resultant, one factor of which will be the primitive function ^ over 
again. There is some reason for supposing that polar reciprocals, which are 
scarcely ever (if ever, except indeed for quadratic functions) the simplest 
contravariants to a given function, may be expressed algebraically by means 
of the simpler contravariants, in the same way as discriminants admit (in 
many, if not in all cases, with the same exception as above) of being repre- 
sented as algebraical functions of invariants of a lower order or simpler 
form. 

I close this section with the remark that every complete and unambiguous 
system of functions of the constants in a given form or set of forms charac- 
teristic* of any singularity absolute or relative in such form or forms must 

* I repeat here that a function or system of functions which severally equated to zero express 
unequivocally and completely the existence of any position or negation, is termed the character- 
istic of such position or negation. Thus for example the resultant of a group of equations 
is the characteristic of the possibility of their coexistence. The discriminant of a function of two 
variables is the characteristic of its possession of two equal factors ; the catalecticant is the 
characteristic of its decomposability into the sum of a defined number of powers of linear 
functions of the variables, &o. 



42] On the Principles of the Calculus of Forms. 305 

constitute an invariantive plexus or set of invariantive plexuses. The system 
unambiguously characteristic of a singularity of an order n will (except when 
n = \) almost universally consist of far more than n functions, subject of course 
to the existence of syzygetic* relations between any ()i + 1) of such functions. 
The existence of multiple roots of a function of two variables is a specific, but 
by no means a peculiar case of singularity, and requires, for its complete and 
systematic elucidation, to be treated in connexion with the general theory 
of the subject. 

Section III. On Commutants. 

The simplest species of commutant is the well-known common deter- 
minant. 

If we combine each of the n letters a, b ... I with each of the other n, 
a, yS . . . X, we obtain n" combinations which may be used to denote the 
terms of a determinant of n lines and columns, as thus : 

aa, a^ ... a\, 

ba, b^...b\, 



la, l^ ... IX. 

It must be well understood that the single letters of either set are mere 
umbrse, or shadows of quantities, and only acquire a real signification when 
one letter of one set is combined with one of the other set. Instead of 
the inconvenient form above written, we may denote the determinant more 
simply by the matrix 

a, b, c ...I, 

a., 13, 7 . . . \ ; 

and to find the expanded value of such a matrix the rule is evidently to take 
one of the lines in all its 1, 2, .3...?i different forms, arising from the 
permutations of the letters (or umbrfe) which it contains ; and then form the 
product of the n quantities formed by the combination of the respective pairs 
of letters in the same vertical column, affecting such product with the sign 
of -H or — according to the rule, that all products corresponding to arrange- 
ments of the terms subject to the permutation derivable from one another 
by an even number of interchanges are of the same, and by an odd number 
of interchanges of a contrary sign. If both lines are permuted and a similar 
rule applied, with the additional circumstance that the sign of the products 

* Rational integer functions which admit of being multiplied severally by other rational 
integer functions such that the sum of the products is identically zero, are said to be " syzygeti- 
cally related." 

S. 20 



306 .On the Principles of the Calculus of Forms. [42 

is made to depend on the product of the algebraical signs due to the respective 
arrangements in the two lines of umbrae, it is evident that the result will be 
the same as when only one line is put into motion, save and except that 
a numerical factor 1 . 2 . 3 . . . ?i will affect each term. If the two sets of umbrse 
a, h, c . . . I \ a, /3, 7 ... /V be taken identical, and if it be convened that the 
order of the combination of any two letters shall not affect the value of the 

quantity thereby denoted, ' ' "' will denote a symmetrical determinant. 

If instead of two lines of umbrse, three or more be taken, the same 
principle of solution will continue to be applicable. Thus, if there be a 
matrix of any even number r of lines each of n umbrse, 

a,, hn ... L, 



a,., br ... Ir, 
the first may be supposed to remain stationary, and the remaining (r — 1) 
lines each be taken in 1, 2 ...n different orders; every order in each line 
will be accompanied by its appropriate sign + or — ; and each different 
grouping in each line will give rise to a particular grouping of the letters 
read off in columns. The value of the commutant expressed by the above 
matrix will therefore consist of the sum of (1.2... ny~^ terms, each term 
being the product of n quantities respectively symbolized by a group of 
r letters and affected with the sign + or — according as the number of 
negative signs in the total of the arrangements of the lines (from the columnar 
reading off of which each such term is derived) is even or odd. 

For example, the value of 

a, b, 

c, d, 
e, f, 
9, h, 
will be found by taking the (1 . 2)' arrangements, as below, 

a, b, a, b, a, b, a, b, a, b, a, b, a, b, a, b, 
c, d, d, c, c, d, d, c, c, d, d, c, c, d, d, c, 
e, f, e, f, /, e, f, e, e, f, e, f, f, e, f, e, 
g, h, g, h, g, h, g, h, h, g, h, g, h, g, h, g. 
The signs of c, c^ ; e,f;g,h being supposed + , those oi d, c; f,e and h, g 
will be each — . Consequently the sum of the terms will be expressed by 
aceg x bdfh — adeg x bcfh — acfg x bdeh + adfg x bceh 
— aceh X bdfg + adeh x bcfg + acfh x bdeg — adfh x bceg. 



42 J On the Principles of the Calculus of Forms. 307 

Commutants thus formed may be termed total commutants, because the 
■entire of each line is made to pass through all its possible forms of arrange- 
ment. In total commutants it is necessary that the number of lines r be 
€ven; for if taken odd, on making all the r lines to change, instead of 
obtaining 1 . 2 . . . n lines, the result obtained when all but one are made 
to change, it will be found that the latter will be repeated ^(1.2...w) 
times with the sign +, and \{\.2 ...n) times with the sign—, so that the 
algebraical sum of the terms will be zero. Moreover the commutants of 
the species above described, besides being total, are simple, inasmuch as all 
the umbrse to be termed consist of single letters. 

My first proposition in the application of the theory of commutants to 
that of forms is as follows : 

Let (^ be a function homogeneous and linear in respect to an even number 
r of any systems whatever of variables, as 

A'l, y^.-.ti] X2, y^.-.ti', Xr, yr-.-tr- 

Form the commutant 

d d d 

dxi ' dy^ '" dti' 

d d d 

dx^ ' dy^ " dt^' 



d d d 
dxj. ' dyr ' ' ' dtr ' 

Let the general term of this commutant, expanded, be called 

Fe^xFe^x ... xFe^, 
then is Si^ei . ^ x Fg^. <j>x ... x Fg^.ij) 

a covariant or invariant*, as the case may be, of (j). 

Be it observed that the march of the substitution for the different sets of 
variables in the above proposition is supposed to be perfectly independent. 
All the systems but one may undergo linear transformation, or they may all 
undergo distinct and disconnected transformations at the same time, and 
the proposition still continue applicable. It will however evidently be no 
less applicable should the march of substitution for any of the systems 
become cogredient or contragredient to that of any other systems. 

If we suppose ^ to be a function of an even degree r of a single system 
of n variables x, y ...t,SQ that the r systems x^, yx, &c., «2, 2/2, &c. ... x^, y^, &c. 
become identical, we can at once infer from the above scheme the existence 
and mode of forming an invariant to (^ of the order n. This last appears 

[* See below, p. 324.] 

20—2 



308 On the Principles of the Calculus of Forms. [42 

for the case n= 2, and ought, for all other values of n, to have been known* 
to the author of the immortal discovery of invariants, termed by him 
hyperdeterminants, in the sense which, according to the nomenclature here 
adopted, would be conveyed by the term hyperdiscriminants. 

Before proceeding to discuss the theory of compound total commutants, 
or enlarging upon that of partial commutants, I shall make an interesting 
application of the preceding general proposition to the discovery of Aronhold's 
S and T, the two invariants respectively of the fourth and sixth orders 
appertaining to a homogeneous cubic function (say F) of three variables 
X, y, z. These may be termed respectively H^ and H^. As to H^ a 
theoretically possible but eminently prolix and ungraceful method im- 
mediately presents itself, namely to take F'^ = Q, and after forming the 
commutant with six lines, 

d d d 
dx' dy' dz' 

d d d 
dx' dy' dz' 

d d d 
dx'- dy' dz' 

d d d 
dx' dy' dz' 

d d d 
dx' dy' dz' 

d d d 
dx' dy' dz' 

to operate with the 6^ ternary products of which this is made up upon 0: the 
result being an invariant of G, will be so to F, and being of the third degree 
in respect to the coefficients of G, will be of the sixth in respect to those of F. 
It will evidently therefore be H^, or at least a numerical multiple of IT^, the 
form of which, inasmuch as the only other invariant is H^, we know in form 
to be unique. But the general theorem affords another and probably the 

* That this was not known explicitly to and should have escaped the penetration of the 
sagacious author of the theory, and those who had studied his papers, must be attributed to the 
imperfection of the notation heretofore employed for denoting the coefficients of a homogeneous 
polynomial function. The umbral method of denoting such a function (j> of the degree r under 
the form of {ax + by + ... + czy, which is equivalent to, but a more compendious and independent 
mode of mentally conceiving and handling the representation 

( d d d\ 

['dx+ydrj+ +^W*' 

exhibits the true internal constitution of such functions, and necessarily leads to the discovery 
of their essential properties and attributes. 



42] On the Principles of the Calculus of Forms. 309 



most practically compendious* solution as regards H^, of ■which the question 
admits. 

Let (rf represent the mixed concomitant to F formed by the bordered 
determinant 



d?F 


d"-F 


d-'F 


dsc- ' 


dxdy' 


dxdz' 


d?F 
dydx 


d'F 

df 


d'^F 
dydz' 


d'F 


d'F 


d^F 


dzdx' 


dzdy' 


dz^' 


I 


V, 


K, 





(? is a function of the second order as to x, y, z, and of the like order in respect 
to ^. Vy ^> which two systems will be respectively cogredient and contra- 
gredient in respect to the x, y, z system in F. In other words, which is all 
we need to look to, (z is a concomitant of F, and so also will be 

G + X{x^ + yv + z^y, 

which may be termed H. Form now the commutant 

d d d 
dx' dy' dz' 

d d d 
dx' dy' dz' 

d d d 
d^' dt]' d^' 

d d d 
d^' drj' d^' 

this being applied to H will give an invariant (the fact that the march 
of the substitutions for the systems x, y, z; ^, rj, ^ is contrary, being com- 
pletely immaterial to the applicability of the general theorem above given) ; 

* Having since this was printed been favoured witli a view of some of the proof-sheets of 
Mr Salmon's most valuable Second Part of his System of Analytical Geometry (about to appear, 
and which is calculated, in my opinion, to awaken a higher idea of and excite a new taste for 
geometrical researches in this country), I iiud that I am mistaken in this point ; the less sym- 
metrical method operated with by Mr Salmon being decidedly the shortest for practically 
obtaining S and T in the general case. Symmetry, like the grace of an eastern robe, has not 
unfrequently to be purchased at the expense of some sacrifice of freedom and rapidity of action. 

t G is the mixed concomitant to the given cubic function, which is halfway (so to speak) 
between it and its polar reciprocal. In fact, when the operation is repeated upon G, which was 
executed upon the given function to obtain G (that is, when we border the Hessian of G in 
respect to x, y, z, vertically and horizontally with the column and line f, ri, f ) the determinant 
thereby represented becomes the polar reciprocal to the given function. 



310 On the Princijyles of the Calculus of Forms. 



[42 



the commutant so formed will be a cubic function of X, in which the coefficient 
of X^ is a numerical quantity, that of X'' is zero, that of X is Hi and the 
constant term is H^. 

Thus for example let i^ = ar* + y' + ^^ + Qinwyz, then 
X, mz, my, 

G = 



mz, 


2/. 


mx. 


V 


my, 


m,x. 


z, 


r 


1. 


V, 


K, 






and therefore 

i? = 2 {(X - m') x^^" + (X + m^) 2yzr}^+ yz^" - 2mx'r]^}, 

the S implying the sum of similar terms with reference to the interchanges 
between x, ^; y, v; z, ^. 

In developing the commutant above, the first line may be kept in a 
fixed position; for the sake of brevity, (x), (y), (z); (f), (tj), (^) may be 
written in the place of 

d d d d d d 
dx ' dy ' dz ' d^ d7} ' d^ ' 

and it will readily be seen that the only effective arrangements will be 
as underwritten : 

i'^) {y) (^) 

{x) {y) {z) 
(V) (0 (?) 
(?) (?) (v) 



(^) (y) (^) 

i^) (^) (2/) 

(?) iv) (r) 

(*) (y) (^) 
{^) (^) (y) 
('?)(?)(?) 
(?)('?)(?) 
(*) iy) (^) 
(2/) (^) (*) 

(?) (?) (^7) 

(?) (?) (»?) 



(«') (2/) (^) 

(«^) (2/) (^) 
(?) iv) (?) 

(?) (^7) (?) 

{^) (y) (^) 
(^) (^) (y) 

(?) (?) (^7) 

(?) (v) (?) 

(«') (2/) (^) 
(*) (^) (y) 
(?) (»?) (?) 
(^7) (?) (?) 
(^) (y) (^) 
(^) («') (y) 
iv) (?) (?) 
('?) (?) (?) 



(«^) (2/) (^) 
(?) (^7) (?) 
(?) iv) (?) 
(?) (v) (?) 

(a^) (2/) (^) 
(^) (a^) (2/) 

(?) (?) (v) 

(?) (?) (^7) 
(x) iy) {z) 

(y) (^) (^) 

(?) (^7) (?) 
(^7) (?) (?) 



(^) (2/) (^) 
(^) (2/) (*•) 
(?) (^7) (?) 
(?) (^7) (?) 

(*) (2/) (^) 
(^) (2/) (*) 
(?) (?) ('?) 
(?) (?) (^7) 

(«) {y) (^) 

(2/) (^) (^) 
('7) (?) (?) 

(?) iv) (?) 



(^) (2/) (^) 
(«') (2/) (^) 
(?) (?) iv) 
iv) (?) (?) 

i^) iy) i^) 
iy) i^) i^) 
(?) (^7) (?) 
(^7) (?) (?) 
(^) iy) i^) 
iy) i^) i^) 
iv) (?) (?) 
(?) (?) (^7) 
i^)iy)i^) 
i^) i^) iy) 
(?) (^7) (?) 
(?) (?) (^7) 



(^) iy) (^) 
iy) i^) i^) 
iv) (?) (?) 
(?) (^7) (?) 
(^) iy) i^) 
(y) («) (^) 
(?) (?) (^7) 
iv) (?) (?) 

(*) (y) (^) 
(^) (^) (y) 
(?) (?) iv) 
(?) (^7) (?) 



42] On the Principles of the Calculus of Forms. 311 

The signs of the four lines in each of these arrangements are two alike, 
and two contrary to the signs of the correspondent lines in the first arrange- 
ment; hence the effective sign is the same for all, and the result, after 
rejecting from each term the common factor — 16, is seen, from inspection, 
to be 

4 (\ - m=)= - Sni^ + 6 (\ - ni^) (X + nv'Y - 12/?i (\ + m=) + 2 (X. + m^ + 1, 
which is equal to 

12X,= + . X= - 12 {m - m") X + 1 - '2Qm^ - 8m' ; 
here the coefficients m — m^ and 1 — 20?7i.' — 8»t'' are the two invariants 
(Aronhold's S and T) for the canonical form operated upon ; and it will 
be observed that 

(1 - 20m' - 8mJ + 64 {m - m'f = (1 + Sm^y, 
which is easily proved to be the discriminant of 
a^ + y'^ + e^ + &mxyz. 

It may however be observed, that this is not the discriminant of the 
function in X just found, as reasons of analogy* might have suggested it 
probably would be : in order that this might be the case, the coefficient 
of X' should be 4 instead of 12, and of X, m — m^ instead of m^ — in. There is 
ground for supposing that another function of X may be found by a different 
method, in which this relation will take effect. 

The theorem above given for simple total commutants admits of an 
interesting application to the general case of a function F of the nth degree, 
in respect to each of two independent systems of two variables x, y; ^, t?. 
Let F be symbolically represented by {ax + hyj^ (a^ + /St/)", so that a"a" 
represents the coefficient of a;"^", na^~%a."' of a;"~'2/p, &c. &c. ; then the 
commutant 

a, b, (1) 

a, b, (2) 

a, b, (n) 

«, A (1) 

«, A (2) 

a, /S, (n) 

will represent a quadratic invariant of F, which will contain (n + 1)^ coefficients. 
By expanding this commutant we obtain a general expression for the invariant 
under a very interesting form. 

* The biquadratic function of x, y having onlj' one parameter, and therefore two invariants, 
its theory possesses striking analogies to the theory of the cubic function of three letters. The 
function in \ which gives these invariants for the first-named function, according to the method 
given in the first section, has the same discriminant as the function itself. 



312 On the Principles of the Calculus of Forms. [42 

I now proceed to give the general theorem for compound total commutants 
as applicable to the discovery of invariants. 

Let there be a function of in disconnected classes of systems of variables ; 
let the systems in the same class be supposed all distinct but cogredient 
with one another. The function is supposed to be linear in respect to each 
system in each class, and the number of systems is the same for all the 
classes, and the number of variables the same in each system. This function 
may then be represented symbolically under the form 

...Ca„.ia;„ + '6„.'2/„+ ...Hn.Hn) 
X {^a^.''x^ + %.-y^ + ... ■r-k.\){'a,..-x, + %.^y^ + ... + % .%) 

... y Ctn • ^n + 'bn • yn + • ■ • "'»i • 'tn) 
X &C. 

X (^Oi .Px, + *6i . ^2/1 + ... +H,.n,) {Pa..Px., + Pb,.Py«_ + ... +H^.n,) 

...{Pan.PXn + ^bn.^yn+---^ln-^k)- 

In this expression the x, y ... t's are all real, but the a, b ... I's all umbral ; 

d d 

in fact, •%£,, /&3, &c. may be understood to denote -jj— , ^j— , &c. 

The n systems of variables in each of the sets above written are supposed 
to be cogredient inter se. 

Take the symbolical product of the first set, first making /or the moment 

^«i = ^^2 = • • • ■'^M = *. &c. &c., Hi = %= ... Hn = t ; 

and let the coefiScients of the several terms 

a;", a;"""^ y ... &c., 

be called ^A^, ^A^ ... 'A,,, 

where //. is the number of terms contained in a homogeneous function of the 
mth degree of the m variables x, y ...t. In like manner proceed with each 
of the lines, and then write down the commutant 

'Ai, 'A,...-'A^, 



pAi, pA,...pA^ 



This commutant is an invariant oi F : it will of course be remembered that, 
unless p is even, the commutant vanishes. 



42] On the Principles of the Calculus of Forms. 313 

Thus, for example, take two sets of two systems of two variables : in all 
four systems, 

^> y'l ^> V '• Pi 1> 4'' 'i''' 

each couple of systems on either side of the colon (:) being cogredient 
inter se : and let F be symbolically represented by 

(ax + by) (a^ + /St?) (Ip + mq) (\(/) + /j-^jr) ; 

then the invariant given by the theorem will be the commutant 

aa ; a^ + ab ; 6/9, 

l\; lfi + Xvi ; mfjL. 

The six positions of this are as below written (the first three being positive 
and the second three negative) 

aa; a/B + ab; b^, aa; a^ + ab; 6/8, aa.; a/3 + a6 ; 6/S, 

IX; i/L(,+\m; mfi, Ifji + Xm; m/j,; IX, mfi; IX; l/j,+ Xm, 

aa; a/3 + ab; bjS, aa; aj3 + ah; 6/3, aa; a^ + ab; bl3, 
Ifj, + Xm ; IX ; nifi, IX ; vifx ; l/j, + Xm, mp, ; Ifi + Xm ; IX. 
If we write F under its explicit form, 

Ax^pcf} + Bx^py^ + Gx^qj) + Bx^q-^ 
+ A'x7]pcf) + B'xrjp-^lr + G'x7]q<^ + D'xrjqyjr 
+ A"y^pj> + B"y^pi^ + G"y^q<^ + I>"y^qy^ 
+ A"'y7]p^ + B"'yrjp-\{r + G"'yriqcf> + D"'yriq-^, 
we have identically the relations following, 

aalX = A, aalfj- = B, aaniX = G, aam/x = D, 
a^lX = A', aj3l/u, = B', a^mX = G', a^nifj, = D', 
balX = A", balfi = B", bamX = G", bamfi = D" , 
b^lX = A"', b/3lfi=:B"', b/3mX = G"', b/Smfj. = D"' , 

and the commutant expanded becomes 

A (B' + G" + G' + B") D'" + (B+G) {D' + D") A'" + D(A' + A") (B'" + G'") 
-(B + G)(A' + A") D'" -A{D' + D") (B'" + G'") -D{B' + G" + G' + B") A'". 

In the foregoing the x's \u. une several lines were for the moment taken 
identical, in order the more easily to explain the law of formation of the 



314 On the Principles of the Calculus of Forms. [42 

quantities A. But suppose that they become actually identical for the same 
line. F then becomes a function of the ?ith degree in respect to each of 
p systems of variables, and may be represented symbolically under the form 

{^a}x + ^6'y + . . . + HHf x (:d'x + ''Py+ ...+ HHY 

... X {PaPx + PhPy + . . . + Hny. 

We may still further limit the generality of the theorem by supposing 

^x = '^x= ...Px = x, 



H = H= ...Pt=t; 

F then becomes {ax+hy+...-\- Ityp. 

Accordingly, as many different factors as can be found contained an even 
number of times in the exponent of the function, so many invariants can be 
formed immediately from a function of any number of variables m by the 
method of total commutation. 

If one of these factors be called n, the commutant corresponding thereto 
will be of the order 

(?i + l)(w + 2)... {n + m-\) 
1.2...(»i-l) 

in respect to the coefficients. Thus take m = 2, so that 

F={ax + hyy'P. 

The general form of such a commutant will be found by taking 
A^, ^2----4n+i tbe coefficients of the several combinations of x, y in 
{ax + hyY, from which the numerical coefficients n, \n {n — 1), &c. may be 
rejected, as only introducing a numerical factor into the result ; the com- 
mutant will therefore be expressed by means of the form 

a"; a^-^h; a"--b- ...; 6», (1) 

a»; a"-^b; a"-=6^...; b«, (2) 



a™; a'^-^b; a''--b- ... ; b\ (p) 

li p = 2, the compound commutant 

a"; a^-'b; ...; 6", 



421 



On the Pt'incijjles of the Calculus of Forms. 315 



will easily be seen to be only another form for the catalecticant of {ax + by)^. 
Thus, let M = 2, 



so that 



(ax + byY ==Ax' + 4>Boi?y + QCa?y- + 4>Dxy^ + Ey^ ; 
a' = A, a>h = B, a''¥=C, a¥ = D, ¥ = E. 



The comrautant (which is of the form of the matrix to an ordinary 
determinant, with the exception that the umbrse enter compoundly instead 
of simply into the several terms separated by the marks of punctuation), 
will be 

d"; ab; b\ 



ab; ¥ 



this, written in the six forms 



a- ; ab 


6n 


a^; ab; ¥ \ a-; ab; ¥'\ 


a- ; ab 


4 


a'' ; ¥; ab \ ab; a^ ; ¥} 


a- ; ab 


¥] 


a-; ab; ¥ ' a^; ab; ¥"\ 


¥; ab 


a^\ 


ab; ¥; a^ j ¥; a''; abj 


gives the expression 




a-> X a-b- x¥-a*x (a¥y- -¥x (a'by - (a^'by + 2a% x ab' x a^¥ ; 


that is J 


WE- 


-AD'-EB'-C' + 2BCD. 



One important observation may here be made of a fact which otherwise 
might easily escape attention, which is, that commutants, where the same 
terms simple or compound are found in all or several of the lines, in general 
give rise to products, some of them equal and with the same sign, and others 
equal but with the contrary sign. 

This last phenomenon does not manifest itself in commutants appertaining 
to functions of two variables of the two particular and different species which 
first and most naturally present themselves, namely where there are only two 
lines or only two columns* — I believe that it displays itself in every other 
case of commutantives to functions of two variables. Thus it is that 
algebraical expressions derived from given functions disguise their symmetry ; 
to make which come to light it becomes necessary to add terms of contrary 
sign to such expressions. As an example, the reader is invited to develope 
the cubic invariant of a function of x and y, symbolically expressed by 
{ax + by)", where 

a"=A, a'b = B ...a¥=R, ¥ = I, 



* These commutants give respectively the quadrinvariant and the catalecticant, the former 
of which alone was formerly recognised by Mr Cayley as a commutant. 



316 On the Principiles of the Calculus of Forms. [42 

by means of the commutant 

a^, ah, ¥, 

a^, ah, ¥, 

a-, ah, h-, 

a^, ah, h".* 

Suppose F to be the general even-degreed function of two variables 
of the degree ^wp. 

Let ^=(^|_,0^i.+ M^?^-3/.)-, 

and express H umbrally under the form 

{ax + hyfP (af + ^-qfP. 

* [See p. 346 below.] The number of terms resulting from the independent permutation 
of each of the 3 linear lines is 6^, that is 216 ; but the actual result is (using small letters instead 
of large) P- Q, where 

P=aei + 3ag"- + 12beh + 3cH + iicp + 2id^g + 15c^ 
Q = iafh + ibid + 8bgf+ llceg + 8chd + 36de/, 
so that the effective number of permutations is only 164. The difference between this and 
216 divided by 216 maybe termed the Index of Demolition, which we see in this case is ^ft-V °^ H; 
that is, somewhat less than J. For the cubic invariant of the function of the fourth degree this 
index is zero, all the permutations being effective. If we take the cubic invariant of the function 
aa;^^ + 126a!^'2/ + 66ca;'''!/- + &c. + m!/'- under the form P- Q, we shall find 

P=6ahl + lOajj + 6bfm + Sibhk + Sicfl + 155c ii + lOddm + iSOdgj 
+ 155eek + o20ehh + 520ffl + 2S0ggg, 

Q = agm + ISaih + SObgl + SObij + IScem + icgh + ISOchj + 30del + 2 lOdfk 
+ 250dhi + 2S0efj + 555egl + GdOfgh. 
The number of terms in P and Q is of course the same, and will be found to be 2200 for each ; 
so that out of the 6^, that is 7776 permutations of the 5 lower rows, only 4400 are effective, and 
the index of demolition becomes f f?-s, that is jjf , or rather greater than yV. The Index of Demo- 
lition thus goes on constantly increasing as the degree of the function rises ; probably (?) it 
converges either towards ^ or else towards unity. In arranging the terms it will be found most 
convenient to adopt, as I have done above, the dictionary method of sequence. The computations 
are greatly facilitated by the circumstance of the effect of any arrangement of each of the 5 lower 
lines not being altered when these lines are permuted with one another ; this gives rise to the 
subdivision of the 7776 permutations into groups as follows : 6 of 120 identical terms, 60 of 60, 
36 of 20, 60 of 80, 24 of 20, 80 of 10, 30 of 5, and 6 of 1. So that the total number of permuta- 
tional arrangements to be constructed is only 252. Other methods of abridging the labour will 
readily suggest themselves to the practical computer. The total number of the groups of terms 
is of course always known a priori, and, for instance, in the case before us, must be equal to the 
number of ways in which J (12 x 3), that is the number 18, can be divided into 3 parts, none of 
which is to exceed the number 12, that is 25 ; for the cubic invariant of the function of the 
eighth degree of two variables it is the number of ways in which 12 can be divided into 3 parts, 
of which none shall exceed 8, and so forth, zeros being always understood to be admissible ; 
and of course in general for an invariant of the order r to a function of the degree n of i variables, 

the number of distinct terms is in general the number of ways in which -r- can be divided into 

r parts, of which none shall exceed n, subject however always to the possibility in particular 
cases of a diminution in consequence of some of the groups assuming zero for their coeflScient. 



a", 


a"-i6 .. 


..h\ 


a", 


a"-'6 .. 


,.i", 


a", 


a"-i& .. 


.&", 


a", 


a^^/3 . 


../3", 


a", 


a"-'^ . 


../3", 



42] On the Principles of the Calculus of Forms. 317 

The commutant 

(1) 
(2) 

{P) 
(1) 
(2) 

a", a"-i/S.../3«, (^) 

will be a function of A., and all the several coefficients will be invariants oiF*. 

When ^ = 1 we obtain the A given in the preceding section, and origin- 
ally published by me in the Philosophical Magazine for the month of 
November, 1851. The A obtained on this supposition has for its coefficients 
a series of independent invariants, commencing with the catalecticant and 
closing with the quadratic invariant. When p has any other value, we 
observe a similar series commencing with a commutantive invariant of a 
lower order than the catalecticant, but always closing with the quadratic 
invariant. Thus, for example, when 2np=S, we may obtain by the preceding 
theorem three different quadratic functions ; one giving the invariants of the 
orders 5, 4, 3, 2, the second those of the orders 3, 2, the third the invariant 
of the order 2. 

In this case the invariants of the same order given by the different A's 
are the same to numerical factors pres. Whether this is always necessarily 
the case is a point reserved for further examination. 

The commutants applied in the preceding theorems have been called 
by me total commutants, because the total of each line of umbrae is permuted 
in every possible manner. If the lines be divided into segments, and the 
permutation be local for each segment instead of extending itself over the 
whole line, we then arrive at the notion of partial commutants, to which 
I have also (in concert with Mr Cayley) given the distinctive name of 
Intermutants. In order to find the invariants of functions of odd degrees, 
the theory of total commutants requires the process of commutation to be 
applied, not immediately to the coefficients of the proposed function, but to 
some derived concomitant form. I became early sensible of this imper- 
fection, and stated to the friend above named, to whom I had previously 

* By substituting the symbols -r- , -7- , &o. in place of the umircs a, b, &c., the theorem 

is easily stated for covariauts generally. But in applying the commutantiTe method to obtain 
covariants, or rather in the statement of the results flowing from each application, it is never 
necessary to go beyond the case of invariants, because the commutantive covariants of any given 
homogeneous function are always identical with commutantive invariants of emanants of the 
same function. 



318 On the Prindjjles of the Calculus of Forms. [42 

imparted my general method of total commutation, my conviction of the 
existence of a qualified or restricted method of permutation, whereby the 
invariants of the cubic function, for instance, of two and of three letters would 
admit, without the aid of a derived form, of being represented. Many months 
ago, when I was engaged in this important research, and had made some 
considerable steps towards the representation of the invariant, that is, the 
discriminant of the cubic function of x and y, under the form of a single 
permutant, I was surprised by a note from the friend above alluded to, 
announcing that he had succeeded in fixing the form of the permutant of 
which I was at that moment in search. It is with no intention of complain- 
ing of this interference on the part of one to whose example and conversation 
I feel so deeply indebted, (and the undisputed author of the theory of 
Invariants,) that I may be permitted to say that, independent of the inter- 
vention of this communication, I must inevitably have succeeded in shaping 
my method so as to furnish the form in question ; and that with greater 
certainty, after my theory of commutants had furnished me with the prece- 
dent of permutable forms giving rise to terms identical in value but affected 
with contrary signs. As I have understood that Mr Cayley is likely to 
develop this part of the subject in the present number of the Journal, it will 
be the less necessary for me to enter at any length into the theory of partial 
commutants on the present occasion. 

The method of partial commutation is a simple but most important 
corollary from that of total commutation hereinbefore explained. To fix 
the ideas, conceive a class of p cogredient systems, and that there are qr 
such classes perfectly independent. Proceed to divide these qr classes in 
any manner whatever into r sets, 'each containing q classes ; and form the 
symbol of the total commutant corresponding to each such set. Now let 
these commutants be placed side by side against one another, and transpose 
the terms in each compound line thus formed once for all, but in any 
arbitrary manner. Then permute in every possible way all those symbols 
in each line, inter se, which belong to the same class, and operate with the 
symbols thus produced by reading off the vertical columns and attending to 
the rule of the -I- and — signs, as in the case of a total commutant ; the 
result will be a commutant of the form operated upon. For instance, let 
p=l, g = 3, r = 2, and let the number of variables in each system be 2. 
Form the commutant operators 



d 


d 


d 


d 


dx' 


dy' 


d^' 


drj' 


d 


d 


d 


d 


dp' 


dt' 


dxt>' 


dd' 


d 


d 


d 


d 


dr' 


ds ' 


dp' 


da-' 



42] On the Principles of the Calculus of Forms. 319 

Interchange in any manner but once for all the symbols in each line, as thus: 
d d d d 
dx' dy' d^' dr}' 

d d d d 
d<l>' dp' dt' dO' 

d d d d 
ds' dp' dr ' da' 

Now permute, inter se, the variables of each system, as 

d d _ d d „ _ 

dx' dy' dp' dt' ' ' 
the total number of the operative forms resulting will be (1 . 2)", and the 
sum of the (1 . 2)" quantities, half positive and half negative, formed after the 
type of 

{d d d jj d d d jj 
dx d(J3 ds dy dp dp 
d d d JJ d d d JJ 
d^ dd dr dt) dt do- 
ll being supposed to be a function homogeneous in 

'^i y ! ^> V ! P< i't </>. ^ ; ^i s ; p, a, 
will be a covariant of U. 

The proof of the truth of this proposition is contained in what is shown 
in the Notes of the Appendix for total commutants, it being only neces- 
sary to make the systems which are independent vary consecutively, and 
then apply the inference to the supposition of their varying simultaneously. 

It may be extended to the more general supposition of classes of an 
unequal number of cogredient systems of unequal numbers of variables 
in each, the only condition apparently required being that the number of 
distinct terms shall be the same in each line of the final commutantive 
operator. The important remark to be made is, that in applying this 
theorem there is nothing to prevent any of the systems being made identical ; 
or, in other words, a given function of one system of variables may be 
regarded as a function of as many different, although coincident, sets as we 
may choose to suppose. Thus, suppose 

U = Ax' + 2Bxy+Cy^ 

we may take the partial commutant formed of the two total commutant 
operators 

d_ d_ 

dx' dy' 

dx ' dy ' 



320 On the Principles of the Calculus of Forms. [42 

combined with itself. If we write them in the same order, 



d 


A 


± 


d 


dx' 


dy' 


dx' 


dy' 


d 


<[ 


d 


d 


dx' 


dy' 


dx' 


dy' 



(where I use the dots and dashes to distinguish those in the same line which 
are considered as belonging to the same class, and therefore as permutable, 
inter se), we shall evidently obtain 4- {AC — B"}- ; if we commence with a 
permutation, so as to have the form of operation 



d d d 


d 


dx' dy' dx' 


dy' 


d d d 


ci 


dx' dy' dx' 


dy' 


it will be found that we obtain 2{AG -B- 


Y-- 


Again, suppose that we have 




U=Ax' + SBx-y + 


Wxf- + Dy' 


If we write 




d d d 
dx' dy' dx' 


d 
dy' 


d d d 
dx' dy' dx' 


d 
dy' 



d d d_ d^ 
dx' dy' dx' dy' 

the value of the commutant would come out zero ; but if we make a permu- 
tation, and write 

d d d d 
dx' dy' dx' dy' 

d d d d_ 

dx' dy' dx' dy' 

d d d d 
dx' dy' dx' dy' 

the operation indicated by the above performed upon JJ, will give a multiple 
of the discriminant of U. 



42] 



On the Princi2iles of the Calculus of Forms. 321 



In like manner we may represent Aronhold's Sextic Invariant of the form 
{x, y, zY by means of the partial commutant 

d d d d d d 

dx' dy' dz ' dx ' dy' dz ' 

d d d d d d 
dx' dy' dz ' dx' dy' dz' 

d d d d d d 

dx' dy' dz' dx' dy' dz' 

If we make 

and use H to signify the determinant 

X, y, z 
I n. % 

which is evidently an universal triple covariant, and make 

and apply to W the partial commutantive symbol 

d d d d d d 
dx ' dy ' dz' dx' dy' dz' 

d d d d d d 
d^' drj' d^' d^' drj' d^' 

d d d d d d 
d^' ' dri' ' d^' ' d^' ' drj' ' d^' ' 

we shall obtain a function of X of which all the odd powers and the second 
power will disappear, and such that the coefficients of X^ and the constant 
term will be Aronhold's S and T, and the discriminant of the entire function 
in respect to X^ (if not for the distribution assigned to the dots and dashes 
in the foregoing, at least for some other distribution) may not improbably be 
the discriminant of the given function (x, y, zf. 

s. 21 



322 On the Principles of the Calculus of Forms. [42 



NOTES IN APPENDIX. 

(1) [p. 295 above.] More generally, in as many ways as the number n 
can be divided into parts, in so many ways can a given function of one set of 
variables be as it were unravelled so as to furnish concomitant forms. 

For instance, the form aa? + Sbx^y + Scxy^ + df has for a concomitant 

aux + buy + hvx + cvy + cwx + dwy, 

where u, v, w are cogredient with x^, 2xy, y'^ ; and also 

auu'x + huu'y + huv'x + hvu'x + cvv'x + cvu'y + cuv'y + dm'y, 

where u, -y; u\ v' are cogredient with each other and with x and y; and the 
proposition in the text may be best derived from this more general theorem 
by dividing the index into equal parts, forming as many systems as there are 
such parts, and then identifying the systems so formed. 

(2) [p. 297 above.] The following additional example will illustrate the 
power of this method. 

Let = {x, y, z)* be the general function of the fourth degree. Form by 
unravehnent the concomitant form {u, v, w, p, q, rf (say P) where u, v, w, p, q, r 
are cogredient with of, y^, 2^ 2s:y, 2xz, 2yx. 

Again, the universal concomitant {x^+yv + ^0^ '^^iH have for its con- 
comitant 

itp + VTj- + tu^- + pri^+ q^^ + r^ri, 

where f , 77, ^ are conti-agredient to x, y, z. Now take the reciprocal polar of 
this last form with respect to ^, 7/, ^; that is, 

1 {vw — Ip") «i^ + 2S {\qr - ^pu) y-^z-^ (say G), 

where x-^, y^, Zi, being contragredient to ^, 17, f, will be cogredient with x, y, z. 
P + XG is a quadratic function of the six variables u, v, w, p, q, r, and its 
discriminant will give a function of X of the sixth degree, all of whose even 
coefficients will be co variants of (f). If we replace x^, y^, z^ by x, y, z, these 
even coefficients will be respectively (understanding that order refers to the 
dimensions quoad the coefficients of (j) and degree to the dimensions quoad 
X, y, z) as follows : 



42] On the Principles of the Calculus of Forms. 323 



Of order 6 


degree 0, 


;. -5 


2, 


„ 4 


4., 


„ 3 


6, 


„ 2 


8, 


,. 1 


„ 10, 


„ 


„ 12. 



The two last coefficients must evidently be identically zero. It is possible 
that some of the others may be so too : as regards the one of the third order 
and sixth degree, this is of the same form as, and may be identical with, the 
Hessian of ^ ; as regards the one of the fourth order and fourth degree, this 
may be <^ itself multiplied by the cubic invariant (which the theory of 
Section III. proves to exist) of ^. But the covariants of the fifth order 
and second degree, and of the second order and eighth degree, if they are 
not identically zero, and if the latter is not c^" (which a trial or two of some 
Tery simple cases will easily establish one way or the other) are probably 
irreducible forms. The existence of a correlated conic section to a curve 
of the fourth order, if established, would be particularly interesting, and its 
geometrical meaning would well deserve being elicited. 

(3) [p. 303 above.] If any form (/) of the degree n be written sym- 
bolically, 

(Oiajj + a^x^ + . . . + a^ajj", 

Avhere x^, x^... x^ are real but a^, aa ... a^ umbral, and if Ir be any invariant 
of the order r in respect of the real coefficients of (/), it is easily seen by 
reason of Ir remaining unaltered when x^, X2 ... x,, become respectively 
_/!«], /jiTa •••/i«o provided that/1,/2 ...f, = \, that each term in /,. expressed 
by means of the umbrae, must contain an equal number of times tti, a^ ... a^, 

so that each such term will contain — of each of them, of course differently 

subdivided and grouped; hence we have the universal condition that — must 
be an integer ; but this is less stringent than the actual condition, which 
is that — must be an integer of a certain form ; for instance, as before 

observed, when t = 2, — must be an even integer. 

(4) [p. 307 above.] To prove the theorem given in the text for total simple 

commutants it is only necessary to bear in mind that whenever two columns 

in any total commutant become identical, the commutant vanishes. To fix 

d d d 
the ideas, take the commutant formed of lines similar to -j- , -j- , j- , written 
' ax dy dz 

21—2 



324 On the Principles of the Calculus of Forms. [42 

under one another; let there be r such lines, the total number of terms 
will be (1 . 2 . 3)'' : the 1.2.3 positions of the line written above will corre- 
spond to (1.2. Kf-'^ several groupings of the remaining lines. Now when 

X, y, z undergo a unimodular linear substitution, ;j~ > j~ = j~ ^'^ undergo 

a related substitution not coincident with that of x, y, z, but still unimodular; 
let X, y, z change, all the other systems remaining fixed, and suppose 

— , T- , -r- to become respectively 
dx dy dz 



.d 


d j^ d 
dy dz' 


/'=+'• 


d J, d 
dy dz' 


ri-^ 


dy dz' 



then each of the (1.2. By~^ groups of the terms arising from the permutation 

of -^ , -T- , -T- will subdivide into 27 groups, of which we may reject those 
dx dy dz 

' d d d' 



in which any of the terms f t- , -5- , -^1 occurs twice or three times ; accord- 

ingly there will be left only the six effective orders of permutations, 

f.d ,d j„ d\ (_fd ,,d „ d's , 
Vdx'^dy'^ dz)' Vdx'^dz'^ dy)' ^'- 

consequently each of the (1.2. Sy~^ groups gives rise to 6 times 6 products 

I /", 9", h" 
whose sum will be /', g', h' x the sum of the 6 products corresponding 

I / 9, h 
to the permutations of t- , t- , -j-; and therefore, the transformations being 

unimodular, the sum of the products corresponding to the entire (1.2.3)'" 
permutations remains constant when x, y, z change. In like manner, all the 
systems may change one after the other, and consequently all of them at the 
same time without affecting the value of the commutant: and in like manner 
for the general case, q.e.d. 

(5) [p. 312 above.] The truth of the proposition relative to compound 
commutants and the mode of the demonstration will be apparent from the 
subjoined example. 

Let the function be supposed to be 

{ax + by) {a'x' + h'y) (a^ 4- ^77) (a'|' -1- /S't?'). 



42] On the Principles of tlie Calculus of Forms. 325 

where x, y; x', y' are cogredient and ^, t}; ^', t) cogredient; the a, &, a, /8, &c. 
are of course mere umbrae. Now take the compound commutant 

aa\ ah' + dh, bb', 
aa', a/3' + a'l3, ^13'. 

Let X, y ; x', y' undergo a linear substitution, and, accordingly, 
let a become fa + gb, 

a' „ fa' + gb', 
b „ ha + kb, 

b' „ ha' + Jcb', 

f, g, h, k being of course actual and not umbral ; then the above commutant 
will be easily seen to decompose into 6 others, which will be equal to the 
original commutant multiplied by the determinant 

f-, ^fg, f 

fh, fk+gh, gk 

If, 2hk, k- 

which is equal to (fk — ghy, that is = 1. 

And so in general, which shows, as in the preceding note, that all the 
classes of cogredient systems may be transformed successively one after 
the other, and therefore simultaneously, without altering the value of the 
commutant. 

(6) In the last May Number* of the Journal, Mr Boole, to whose modest 
labours the subject is perhaps at least as much indebted as to any one other 
writer, has given a theorem f, (14) p. 94, the excellent idea contained in 
which there is no difficulty in shaping so as to render it generalizable by aid 
of the theory of contravariants. It may be regarded in some sort a pendant 
or reciprocal to the Eisenstein-Hermite theorem, presented by me under a 
wider aspect in the First Section of this paper. 

[* Gamb. and Dub. Math. Journ. Vol. vi. (1851), pp. 87—106.] 

+ Mr Boole applied his theorem to obtain the cubic invariant of {x,y)*, say ^{x,y), by 

operating upon its Hessian with 0(-r-, - -r- 1 • More generally, when 4i{x, y} = {x, yf'^, the 
\dy ax J 

catalecticant of the antepenultimate emanant of <p is also of the degree 2n ; and this, when 

operated upon by ( — , - -;- | , will give an invariant of the order m + l, which is probably 
\dy dxj 

identical with the catalecticant of itself. There exists a most interesting transformation of the 

catalecticant of any emanant of a function of any degree in x, y, whether even or odd, under the 

form of a determinant some of the lines of which contain combinations only of x and y, without 

any of the coefficients, and all the rest the coefficients only of the given function without x or y. 

The Hessian being the catalecticant of the second emanant is of course included within this 

statement. 



326 On the Princijyles of the Calculus of Forms. [42 

Let (f>(x, y ... z) have any contravariant (x, y ... z); then will 

be a contravariant of ^. For orthogonal transformations the terms contra- 
variant and covariant coincide, and the theorem for this case appears to have 
been known to Mr Boole, see (15), same page. More generally, if -v/r and 9 
be any two concomitants of ^, the algebraical product ■y^d will also be a 
concomitant of ^, provided that the systems of variables in '>^ and 6 have all 
distinct names, or that those which bear the same names are cogredient with 
one another. If this proviso does not hold good, the product in question will 
evidently be no longer a concomitant of 4>. Let however ^ denote what -v/r 
becomes, and ^ what 6 becomes, when in place of the variables x, y ... z 
of every two contragredient synonymous systems in -^ and Q we write 

-- , -T- -=- , then will ^tV and "^0 be each of them concomitants of 6, 

ax ay . dz 

the synonymous systems becoming cogredient with i/r in the one case and 

with 6 in the other. 

(7) There is one principle of paramount importance which has not been 
touched upon in the preceding pages, which I am very far from supposing 
to exhaust the fundamental conceptions of the subject, (indeed, not to name 
other points of enquiry, I have reason to suppose that the idea of contra- 
gredience itself admits of indefinite extension through the medium of the 
reciprocal properties of commutants ; the particular kind of contragredience 
hereinbefore considered having reference to the reciprocal properties of 
ordinary determinants only). 

The principle now in question consists in introducing the idea of con- 
tinuous or infinitesimal variation into the theory. To fix the ideas, suppose 
to be a function of the coefficients of t^ {x, y, z), such that it remains 
unaltered when x, y, z become respectively /«, gy, hz, provided ^&t fgh = \. 
Next, suppose that G does not alter when x becomes x-\-ey + ez, when e and 
e are indefinitely small : it is easily and obviously demonstrable that if this 
be true for e and e indefinitely small, it must be true for all values of e and e. 
Again, suppose that C alters neither when x receives such an infinitesimal 
increment, y and z remaining constant, nor when y nor z separately receive 
corresponding increments, z, x and x, y in the respective cases remaining 
constant ; it then follows from what has been stated above that this remains 
true for finite increments to a; or y or z separately ; and hence it may easily 
be shown that C will remain constant for any concurrent linear transforma- 
tions of X, y, z, when the modulus is unity. This all-important principle 
enables us at once to fix the form of the symmetrical functions of the roots 

of ^ (- , 1 ) which represent invariants of (j> (x, y) when the coefficient of the 



42] On the Principles of the Calculus of Forms. 327 

highest power of x is made unity. It also instantaneously gives the neces- 
sary and sufficient conditions to which an invariant of any given order of any 
homogeneous function whatever is subject, and thereby reduces the problem 
of discovering invariants to a definite form. But as these conditions coincide 
with those which have been stated to me as derived from other considerations 
by the gentleman whose labours in this department are concomitant with my 
own, I feel myself bound to abstain from pressing my conclusions until he 
has given his results to the press. 

(8) By aid of the general principle enunciated in Note (6) above, we caa 
easily obtain Aronhold's S and T. Let U be the given cubic function of 
X, y, z, and let (x, y, z; ^, t), t,) be the polar reciprocal in respect to ^, ??, % 

(d d d\^ 
^ j~ + V-j- + ^-i-) ^> then G {^, rj, ^; x,y,z) as well as the former Q 

will be a concomitant to U, but the homonymous systems of variables in the 
two G's will be contragredient ; and, accordingly, 

^(d d d d d d\ ^ 
^Xd^'Ty'dz' WTv'W-^^^''^'^'''''y''^ 

will be a concomitant to JJ ; this concomitant is readily seen to be an 
invariant of the fourth order; that is, Aronhold's S. Again, from ;Sf, by 
means of the Eisenstein-Hermite theorem, we may derive a form K (x, y, z) 
of the third degree in x, y, z, and whose coefficients will be of three dimen- 
sions ; and, accordingly, if the Hessian of U be called H ( U), 

will be a Sextic Invariant of JJ, that is, Aronhold's T. 



43. 

ON THE PRINCIPLES OF THE CALCULUS OF FORMS. 

[Cambridge and Lublin Mathematical Journal, vii. (1852), pp. 179 — 217.] 

Part I. Section IV. Reciprocity, also Properties and Analogies 
of certain Invariants, &c. 

It will hereafter be found extremely convenient to represent all systems 
of variables cogredient with the original , system in the primitive form by 
letters of the Roman, and all contragredient systems by letters of the Greek 
alphabet; the rules for concomitance may then be applied without paying 
any regard to the distinction between the direction of the march of the 
substitutions, the variables at the close of each operation as it were telling 
their own tale in respect of being cogi-edients or contragredients. This 
distinction has not (as it should have) been uniformly observed in the 
preceding sections ; as, for instance, in the notation for emanants which have 

been derived by the application of the symbol ( I ;7~ + ^7 t~ + &c. j , instead 
of the more appropriate one W -j- + y' t- + &c. 



dx '' dy 

The observations in this section will refer exclusively to points of doctrine 
which have been started in the preceding sections in such order as they more 
readily happen to present themselves. And, first, as to some important 
applications of the reciprocity method referred to in Notes (6) and (8) of the 
Appendix [pp. 325, 327 above]. 

The practical application of this method will be found greatly facilitated 
by the rule that x, y, z, &c. may always in any combination of concomitants 

be replaced respectively by -jz , -^ , tt, , &c., and vice versa. I shall apply 

this prolific principle of reciprocity to elucidate some of the properties and 
relations of Aronhold's >S and T, and certain other kindred forms. This 
/S and T are the quartinvariant and sextinvariant respectively of a cubic 
of three variables. I give the names of s and t to the quadrinvariant and 
cubinvariant of the quartic function of two variables. Furthermore, whoever 
will consider attentively the remarks made in Section II. of the foregoing 
relative to reciprocal polars, will apprehend without any difiSculty that to 
every invariant of a function of any degree of any number of variables will 



43] 071 the Principles of the Calculus of Forms. 329 

correspond a contravariant of a function of the same degree of variables 
one more in number, and that between such invariants, whatever relations 
exist expressed independently of all other quantities, precisely the same 
relations must exist between the corresponding contravariants. Thus, then, 
to s and t the two invariants of {x, y)* will correspond two contravariants 
<7 and T of (x, y, zf, and to B and T the two invariants of {x, y, zf will 
correspond 2 and S- two contravariants of (a;, y, z, if. Calling r the resultant 
of (x, yY, R the resultant of (,«, y, zf, p the polar reciprocal, or, more briefly, 
the reciprocant of {x, y, z)*, and (i?) the reciprocant of {x, y, z, ty, we have 
the following equations (presuming that all the quantities are previously 
affected with the proper numerical multipliers), namely 

r^s' + t", p = a'^- + T-, 

R=S' + T\ (i?) = S' + ^=. 

I propose in this First Annotation to point out the remarkable analogies 
which exist between the modes of generating the four pairs of quantities 
s, t, &c., the functions severally corresponding to which I shall call u, a, U, Q,. 
The Hessian corresponding to any of these functions Avill be denoted by 
an If prefixed, and when we have to consider, not the pure Hessian, but the 
matrix formed from it by adding a vertical and horizontal border of variables, 
the same in number but contragredient to the variable of the function 
(as, for instance, the Hessian of v, bordered with ^, r) horizontally and verti- 
cally, or of U with ^, 77, ^), then I shall denote the result by the ruled 
symbol H, and if there be occasion to add two borders, as ^,77, f ; ^', 7;', f', 
both repeated in the horizontal and vertical directions, the result will be 
typified by the doubly ruled H. 

Now, in the first place, as observed by me in Note (8) of the Appendix 
in the last number; if we call the coefficients of U (10 in number) a, b, c, d, 
&c., we have 

„ fj.( d d d d d d] ^. ^, 

^ = ^te' dv' d-r dx' dy' cT.r''^' ^' "'' ^' ^' ^^' 
also 

dSd'R dS d'H dS dm 

da dcc^ db d'^xdy do d^xdz 

I will now add the further important relation 

d^d^ dT d'H dT d'H ^^^ 
da da? db d-xdy do d-xdz 

* It will be found hereafter convenient to designate contravariants formed in this manner 
from invariants as Evects of such invariants or contravariants, and according to the number of 
times that such process of derivation is applied, 1st, 2nd, 3rd, &c. evects. Such evects form 
a peculiar class, and when considered generally, without reference to the base to which they 
refer, they may be termed evectants. Evectants will be again distinguishable according as their 
base is an invariant simply or a contravariant. Perhaps the terms pure and affected evectants 
may serve to mark this distinction. 



330 On the Principles of the Calculus of Forms. [43 

so that it will be observed if all the derivatives of S are zero, T is zero; 
and vice versa. 

Precisely in the same way, using h and h to denote respectively the 
Hessian of u and the same bordered with ^, 77, we have 

ds d^h ds d^h ds d^li 

da daf dbdar'dy dcda?dy- 
2 _ dt d*h dt d*h dt d^h „ 

da daf db da?dy dc dw-dy^ 
Again, taking (H) the second bordered Hessian of O; that is, D, bordered 
as well horizontally as vertically with the double lines and columns ^, t], ^, d ; 

r> v, r> 0\ 

^ _(Tj\( d d d d d d d ^ yi ' y' q\ 
^~^ ^[dl' Ir]' d^'_dd' di' d^' di' dt' ^' '^' ^' ) 

x(f)(^, y,z,t; I ■n,^,e; r, V, r, e'\ 

^_dld'H dl d'H dl d'H d%_dm 

da da? db dx'dy dc dx-dz dd da?dt '' 

da da? db dx^dy 
In like manner again 



,r^ { d d d 


d d d 

dx' dy' dz' 


^_dad'(h) 

da dx* ■' 


X h {x, y, z 


2 d-^ d'h _ „ 
da dx^ 





r, '?', rj 
^,v,K; r. v\ n 



o" and T are the same quantities as are calculated by Mr Salmon, in his 
inestimable work On Higher Plane Curves, but are there expressed under 
the names of S and T, with the sole difference that in place of x, y, z, used 
by Mr Salmon, the contragredient variables f ', 97', l^' are used in the expressions 
above. Mr Salmon has also pointed out to me that o- may be obtained 
by operating with 

directly upon / a cubic invariant of the function m, or {x, y, zy. This 
I is no other than the simple commutant obtained by operating upon u 
with the commutantive symbol formed by taking four times over the line 

3- , -^ , -r , agreeable to the remark made in the third section that 
dx ay dz ° 



43] On the Principles of the Calculus of Forms. 331 

every function of an even degree of n variables possesses an invariant of 
the nth order in extension of Mr Cayley's observation that every such function 
of two variables possesses a quadrin variant, that is an invariant of the second 
order. 

I need hardly remark that cr is of 2 dimensions in the coefficients and of 
4 in the contragredient variables, t of 3 in the coefficients and of 5 in the 
contragredients, S of 4 in the constants and 4 in the contragredients, ^ of 6 
in the constants and 6 in the contragredients, or that the single-bordered 
Hessians of u and U and the double-bordered Hessians of w and SI are each 
of them quadratic in respect of the x &c. as well as of the ^ &c. systems. 

If the right numerical factors be attributed to S, T, Aronhold has shown 
that 

H[H{V)] + T.H{U) + 8'V = Q, 

and in my paper in the last May Number*, I gave the equation 

h {h (m)| +s.h (u) + tu = 0. 
I think it highly probable that it will be found that the analogous equations 
obtain, namely _ _ _ 

H {H{D,)] -H ^ . jy (O) -f 2"-0 = 0, 

h {Ti {(o)] + cr .h{(o) + To> = 0. 
These remarkable equations, if verified (of which I can scarcely doubt), will be 
most powerful aids to the dissection of the forms a>, D,, and thereby to the 
detection of the fundamental properties of curves of the fourth and surfaces 
of the third degree, of which at present so little is known. It will have been 
observed that in the preceding developments the contravariants of w and O 
were derived in precisely the same way from (o and £1 as the corresponding 
invariants of u and JJ from u and JJ, with the sole difference that the Hessian 
used in the two latter cases is replaced by a single-bordered Hessian in the 
two former cases, and a single-bordered Hessian in the two latter by a 
double-bordered Hessian in the two former. The analogies are not even yet 
stated exhaustively ; for it will be remembered (as shown in the third section), 
that J" and iS can be derived directly and concurrently by means of operating 
with the commutantive symbol 

d d d' 
dx' dy' dz 

d d d 
dx ' dy' dz 

d d d 
T^' d^' ^1 

c^ d d\ 
d^' dr)' d^j 

I* p. 192 above.] 



upon H(U) + \{x^ + yv+ ^?)^ 



332 On the Principles of the Calculus of Forms. [43 

which gives a result of the form m (X^ + SX + T), m being a number ; and 
I conjecture that if 

d d d d 
dx ' dy' dz' dt' 

d d d d 

dx' dy' dz' dt' 

d d d d 
5|' d^' d~^' dd' 

d d d d 
d^' d^' T^' dd' 
be made to operate upon 

Fn + X {x^ +yr] + z^+ tey, 

and the result be put under the form 

ni(\* + AX' + B\^ + Ox + D), 
that A will be zero, B and C will be respectively 2 and ^, and perhaps 
D (a contravariant, if it effectivelj^ exist, of 8 dimensions in the coefficients 
of 12, and of a like number in the contragredients f', v\ ?' . ^')' ^1^° zero. 
But of the evanescence of Z) I do not speak with any degree of assurance. 

Mr Salmon has made an excellent observation to the effect that if we call 

d d d 
(o-) what a- becomes when ^', tj', ^' are replaced hj -^ , -t- , -r, {a-)h (ta) 

will represent a covariant to w of 3 + 2, that is, 5 dimensions in the coefficients, 
and of 6 — 4, that is, of 2 dimensions in x, y, z, h (to) being of 3 and 6 dimen- 
sions in these respectively, and o- of 2 and 4 dimensions respectively in the 
same. Now these resulting dimensions 5 and 2 precisely agree with the 
form especially noticed by me in Note* (2) of the Appendix, where it was 
derived as one of a group by the method of unravelment. There can 
be little doubt that these two conies each of them indissolubly connected 
with every curve of the fourth degree are identical. The form (cr)h{(o) 
enables us to prove readily (thanks to Mr Salmon's calculation of a, given 
in his Higher Plane Curves, under the name of S) that this is a bond fide 
existent conic. 



we find 



For if we take a particular case of to, say 

o) = a^x^ + br,y* + c-jZ* + Qdy-z-, 



h (a>) = aiof, 0, 

0, Ihy" + dz", dyz 
0, dyz, c^z- + dy- 

= (Xi (62C3 + d-) x-y-z- + Orjiodx^y* + Orfi^dx^z^, 
[* p. 323 above.] 



'^'iiJ 



43] On the Principles of the Calculus of Forms. 333 

and (T becomes 

Oidr]'-^'-, 

and consequently (a-) is 

\dyj \dzj 
and therefore 

(o-) h (tu) = 'iai-d (&2C3 + c^-) a^, 

the conic here reducing to a pair of coincident straight lines. This example 
demonstrates that the conic is in general actually existent. 

As I have said so much upon S and T it may not be irrelevant to state 
in this place how I obtained the conditions for U, the characteristic of 
the curve of the third degree becoming the characteristic of a conic and 
a straight line, that is breaking up into a linear and a quadratic factor, 
which Mr Salmon has inserted in the notes to his work above referred 
to. When U is of this form it may obviously by linear transformations be 
expressed by ax^ + Qdxyz, but when starting with the general form, 

a^a? + h^^ + c^z^ + kiO.. + &Dxyz, 

we form two contravariants from S and T, to wit 

and then make 0^ = a, D = d, and all the other coefficients zero, it will easily 
be seen on examining the forms of S and T, given by Mr Salmon, that (S) 
and (T) (the evectants of <Si and T) become respectively 

we have therefore (T) + X (S) = : and (T) and (S), although contravariantive 
to their primitive U, are covariantive with one another, so that (T) + X (S) = 
is a persistent relation unaffected by linear transformations ; it follows 
therefore that when U is of, or reducible to, the form supposed, 

dS . dS . dS dS 

doi ' dhi ' dcs ' ' ' dD 

^dT .dT dT dT 

da^ ' db.2 ' dCi ' ' ' dD ' 

which is the criterion given in the note referred to*. 

I am also able to obtain these equations more directly by another method 
founded upon a New View of the Theory of Elimination, an account of which, 

* Mr Salmon has remarked that the two evectants (S) and (T) intersect in the nine cuspidal 
points of the polar reciprocal to the curve. 



334 On the Principles of the Calculus of Forms. 



[43 



however, I must reserve for another occasion, but which, I may mention, 
serves to fix not merely the conditions, as in the ordinary restricted theory, 
that a given set of equations may be simultaneously satisfiable by some one 
system of values of the variables, but the conditions that such set of equations 
may be simultaneously satisfiable by any given number of distinct systems 
of variables. 

Mr Salmon has remarked to me to the effect that if in t we write 

-^, -— , -r , in place of the contragredients, and call t so altered (t), 
ax ay dz ^ 

then (t)A((») will be an invariant of 6 dimensions in the coefficients of to. 

This sextinvariant I have little doubt is identical with that obtained by 

operating upon &> with the commutantive symbol 

f d\- d^ d^ f dV- d^d^ /d\- d^ d\ 
dx dy ' \dyj ' dy dz' \dz) ' dz dx\ 

d d f d\- d d fd\- d d 



Kdx) 



dx, 



dz) ' dz dx) 



dx dy ' \dyj ' dy dz' 

This, like every other commutant of 2 lines only, is of course capable of being 
expressed under the form of an ordinary determinant, and the remark is not 
without interest, as showing how the proposition known with respect to 
quadratic functions of any number of variables, namely of every such having 
an invariantive determinant, lends itself to the general case of functions 
of any even degree of any number of variables which also have always an 
invariantive determinant attached to them, of which the terms are simple 
coefficients of such functions. The only peculiarity (if it be one) of quadratic 
functions in this respect being that they have each but one invariant of such 
form and no other. In the case before us, if we write 

w = tti*'' + ia^* + Cs^ + M^x^y + 4>asxr'z + ib^y^x + ^ib^y^z + ^c-^z^x + ^c^^y 
4- 6dy-z- + 6ezV + dfafy" + I2lx-yz + 12mxy^z + 12nxyz^, 

the sextinvariant in question becomes representable under the form of 
the determinant 

«i> a«, f, I, e, ttj 

fits I f, bi, in, n, I 

f, bi, bo, bs, d, m 

I, in, 63, d, Co, n 

e, n, d, C2, C3, Ci 

da, i, in, n, Ci, e 

* This determinant is identical with the determinant formed by taking the second differential 
coefficients of the function and arranging in the usual manner the coefficients of the several 
powers and combinations of powers of the variables treated as if they were independent quantities. 



43] On the Principles of the Calculus of Forms. 335 

Before quitting the subject of S and T the two invariants of the cubic 
function of 3 variables, or, as it may be termed, of the cubic curve, it may 
not be amiss to give the complete table which I have formed corresponding 
to all the singular cases which can befall such curve, which will be seen below 
to be eight in number ; it is of the highest importance to push forward the 
advanced posts of geometry, and for this purpose to obtain the same kind 
of absolute power and authority over, and clear and absolute knowledge of, 
the properties and affections of cubic forms as have been already attained for 
forms of the second degree. 

Let U = a*' + ^shoc-y + icx-z + &c. 

(1) When U has one double point 8' + !"-=^ 0. 

(2) When U has two double points, that is becomes a conic and 
right line 

dSdT_dSdT^^ ^^ ^ 
da db db da ' ' ' 

(3) When U has a cusp 8=0, T = 0. 

(4) When U has two coincident double points, that is, is a conic 
and a tangent line thereto, which comprises the two preceding cases in one, 

-J— = 0, -TT = Oi &C. 

da db 

and also therefore 8 = 0. 

(5) When U becomes three right lines forming a triangle 

d-S d'T _ _^ d'8 _ ^^ 
dadb dcde dadb dcde 

where a, b, c, e each represent any of the coefficients arbitrarily chosen, 
■whether distinct or identical. 

Another, and lower in degree system of equations, may be substituted 
for the above, obtained by affirming the equality of the ratios between 
the coefficients of U and the corresponding coefficients of its Hessian. 

(6) When U represents a pencil of three rays meeting in a point 

d8 .. dS ,, (, 
^- = 0, ,,- = 0, &c. 
da db 

and also therefore T =0. 

Also in place of this system may be substituted the system obtained by 
taking all the coefficients of the Hessian zero. 



336 On the Principles of the Calculus of Forms. [43 

(7) When U becomes a line, and two other coincident lines, 

T- = 0, -TT = 0, &c. 

da do 

A 1 d"T .. d'T . . 

and also ^— = 0, -, — r- = 0, &c. 

dcv dadb 

I have not ascertained whether this second system necessarily implies the 
first; I rather think that it does not. In the preceding case also it would 
be interesting to show the direct algebraical connexion between the system 
formed by the coefficients of the Hessian and the system consisting of the 
first derivatives of S. 

(8) When U becomes a perfect cube representing three coincident 
right lines 

d'S . d'S . „ 

T-r = 0; -1 iT = 0, &C. 

da^ dado 

, d'T ^ d'T . . 

and T— „ = 0, -j—TT = 0, &c. 

da- dado 

The first of these systems of equations necessarily implies the equations 

dT dT 

—- = Q, -j^ = 0, &c., as is obvious from the equation 
da do 

dSdPH dS d'H ^^ 

da dx^ db da?dy 

d^T 
but not necessarily the second and lower system -^ = 0, &c. above written. 

So if we take 

v, = asc^ + ^hx^y + Qcoc^y- + Mxy^ + ey* 

when 2 roots are equal 

s' + t- = 0, 

when 2 pairs of roots are equal 

ds dt ds dt _^ „ 
da db db da ' '' 

when 3 roots are equal s = 0, i = 0, 

and when all 4 roots are equal 

dt ^ dt ^ ^ 
-r- = 0, TT = 0, &c. 
da do 

Before closing this Section I may make a remark, in reference to the sextic 
invariant of a, which admits of being extended to all commutants formed 
by operating upon the function with a commutantive symbol obtained by 

writing over one another lines consisting of powers of t-, -t-, &c. and 



43] On the Prmciples of the Calculus of Forms. 337 

their combinations (to which, in the Third Section, I gave the name of 
compound commutants, a qualification which, for reasons that will hereafter 
be adduced, I think it advisable to withdraw). The remark I have to make 
is this, namely that the invariant obtained by operating upon w with 

d\- d d I dy d d / dV d d\ 



■ a 



dxj ' dx dy' \dyj ' dy dz' \dzj ' dz dx\ 

_^Y d d / dy d d /d_\'' ^ <^ \ 
\dx) ' dx dy' \dyj ' dy dz' \dzj ' dz dxi 

is precisely the same as may be obtained by operating with 

d d d d d d\ 
du' dv' dw ' dp' dq' dr\ 

d d d d d d i 
du' dv' dw' dp' dq' dri 

upon the concomitant quadratic function to m obtained by the method of un- 
ravelment, as in Note (2) of the Appendix [p. 322 above] ; and so, in general, 
every commutant obtained by operating upon a function of any number 
of variables of the degree 2mp with a symbol consisting of 2p lines in which 

the mth. powers of ^- , t- , &c. and their mth combinations occur, will 
dx dy 

be identical with the commutant obtained by operating with a symbol 

also of 2p lines, in which only the simple powers occur of -j- , -j- , &c. 

(where u, v, &c. are cogredient with x^, x^-^y, &c.), upon a function of 
M, V, &c., formed by the method of unravelment from the given function. 

Finally, before quitting the subject of reciprocity, I may state, it follows 
from the general statement made at the commencement of this Section, that 
inasmuch as 

{x^ + y7) + z^ + &c.y 

is a universal concomitant form, so also must 

/ d d d d d d „y 
\d^ dx dr) dy d^ dz ' ) 

be a universal concomitant symbol of operation ; accordingly it is certain 
that any concomitant in which x, y, z, &c., f, rj, f, lic. enter, operated 
upon with this symbol, will remain a concomitant: in several cases which 
I have examined, the effect of this operation will be to produce an evanescent 
form, but I see no ground for supposing that this is other than an accidental, 
or at all events for supposing that it is a necessary and universal consequence 
of the operation. It may also be observed that in the case of as many 
cogredient sets of variables as variables irt each set, as for instance 3 sets 
s. 22 



338 On the Principles of the Calculus of Forms. [43 

of 3 variables each, the determinant which may be formed by arranging them 
in regular order, as 

«. y, 
«'> y, 
«", y" 

is evidently a universal concomitant, and moreover an equivocal concomitant, 
possessing the property of remaining a concomitant when the variables 
are respectively but simultaneously exchanged for their contragredients 
^. V) K'l ^'. '?'. ?'j ?"> v. ?"> which shows also that in place of the variables 
may be written the differential operators 

d d d d d d d d d 

dx' dy' dz' dx' ' dy' ' dz' ' dx" ' dy" ' dz" ' 

a remark which leads us to see the exact place in the general theory occupied 
by Mr Cayley's method of generating covariants given in the concluding 
paragraph of the First Section [p. 290 above]. I may likewise add, that 
inasmuch as {x^ + y'rj + z'^ + &c.)^ is a universal concomitant, 

, d , d , Y 
'dx+^ Ty + H 

will be so too, by virtue of the general law of interchange, which conducts 
immediately to the theory of emanation, showing that this last symbol, 
operating upon any function, furnishes covariants thereunto for any integer 
value of z. 

One additional interesting remark presents itself to be made concerning 
U, the cubic function of x, y, z, which is, that calling as before T its sextic 
invariant, and a, 36, 3c, d, &c. the coefficients, the formula 

1^ ^. d ^„,. d ^ ,, d 



(f|.+f-'s+f'fi+ffB+'«^ 



will give the polar reciprocal, or, as it has been agreed to term it, the 
reciprocant of U. I believe the remark of the probability of this being 
the case originated with myself but Mr Cayley first verified it by actual 
calculation, using for that purpose the value of T, given by Mr Salmon 
in his work On the Higher Plane Curves, already frequently alluded to, 
which is an indispensable manual equally for the objects of the higher 
special geometry as for the new or universal algebra, being in fact a common 
ground where the two sciences meet and render mutual aid. 

Mr Salmon also observed, that the first evect of T, namely 



43] On the Principles of the Calculus of Forms. 339 

was identical in form with what may be termed the first devect of the polar 
reciprocal, that is, the result of operating upon the polar reciprocal with 

what TJ becomes when -rz., -t-, -n^, are substituted in the stead of x, y, z. 
d^ 0,7) d^ •' 

And inasmuch as, by Euler's law, 

it follows that T is the second devect of the polar reciprocal, or at least 
identical with it in point of form. But, since the preceding matter was 
printed, I have discovered in the course of a most instructive and suggestive 
correspondence with Mr Salmon, the principle upon which these and similar 
identifications depend, thereby dispensing with the necessity for the exces- 
sively tedious labour of verification which, even in the simple example before 
us, would be found to extend over several pages of work. 

The theory in which this principle is involved will be given, along with 
other very important matter, in the next number of the Journal. 

Supplementary Observations on the Method of Reciprocity. 
It has been observed, that ^, 17, &c. may always be inserted in place of 
-J- , T- , &c., and vice versa, in a concomitant form, without destroying 
its concomitance. Accordingly, instead of the evector symbol 



we may employ 



fE+f^®+^'=- 



dxj da \dx, dy db '' 



and operating with this upon any concomitant, the result will be a concomitant. 
Hence we see, for example, that if we take the concomitant SH formed 
by the product of the invariant S and the covariant H, 

^dxj da \dx) dy db '] 

will be a covariant ; in fact this will be found to be T, the difference 
between this and the expression before given for T, namely 



being 



(d_,^TTdS /dy d dS „ 
\dxJ da \dx) dy db '' 

\da \dx) db \dx) dy 

22—2 



340 On the Principles of the Calculus of Forms. [43 

which is zero, there being no invariant to {x, y, zf of the 3rd degree in 
a, b, c, &c., as the factor multiplied by /S would be were it not evanescent. 
The same observation may be extended to analogous equations given 
previously. 

I have chiefly, however, made the above observation with a view to 
making more clear the enunciation of the theorem which I am now about 
to state, the most important perhaps in its application of any yet brought 
to light on the subject, but the consequences of which, as I have but quite 
recently discovered it, must be reserved for a future number of the Journal. 

Let any function of any number of variables be supposed to have for 
its coefficients the letters a, b, &c. affected with the ordinary binomial or 
multinomial coefficients; and let another function be taken identical with 
the former in all respects, except in the circumstance that all their numerical 
multipliers are suppressed. Let this function or form be termed the respondent 
to the primitive : furthermore, by the inverse of any form understand what 
that form becomes when, in place of x, y, z, &c., ^, r), f, &c., 

d d d „ d d d J, 

dx' dy' dz' '' d|' dr]' d^' '' 

are respectively substituted (and so for all the systems of the variables), and 

likewise at the same time similar substitutions are made of -^, -^ , ^-, &c., 

da db dc 

in place of a, b, c, &c. ; then we have this grand and simple law — The inverse 

of any concomitant to a respondent is a concomitant to its primitive. When 

the inverse of any concomitant to the respondent is made to operate upon 

the same concomitant of the primitive, it will be found that the result 

is a power of the universal concomitant. If the concomitant to the respondent 

be an invariant thereof, the rule indicates that on merely replacing in the 

respondent a, b, c, &c. by ^ , -tt, -r , &c., the result operating on any 

invariant or other concomitant of the primitive, leaves it still an invariant 
or other concomitant. For instance, if we take the function 

aaf + bbsd^y + \Qca?y'^ + lOda^y'^ + bexy* +fy\ 

which has three invariants L, M, N, of the degrees 4, 8, 12, respectively : 
and if we call X, jj., v what L, M, N become when, in place of a, b, c, d, ef 
respectively, we write 

A. lA Li. LA. I A A 

da' 5 db' 10 dc' 10 dd' ode' df 
we shall find that 

and 

XN = a linear function of M and L^.. 



43] On the Principles of the Calculus of Forms. 341 

Again, if in the case of any function of os, y, z, &c., we take, instead of any 
other concomitant to the I'espoudent, the respondent itself, its inverse gives 
the symbol of operation 

\daj \dxj db \dxj \dyj '' 

just previously treated of If again, in the case of a function of x, y, say 

ax^ + nhx''^~^y + . . . + nh'xy'^-'^ + a'y'^, 

we take the inverse of the polar reciprocal of the respondent, we get the 
operator 

A (^AV _ A (AX~'' — 4- & • 
da \drjj db \dr]J d^ ' ' 

and replacing -y- , jz^y y, *', we find that 

yd^-y''db+^''-' 

operating on any concomitant, leaves it still a concomitant, which is 
M. Eisenstein's theorem before adverted to, only generalized by the in- 
troduction of any concomitant in lieu of the discriminant. 

This extraordinary theorem of respondence will be found on reflection 
to favour the notion of treating the coefficients of a general function as 
themselves a system of variables, in a manner contragredient to the terms 
to which they are affixed. 

Finally, there is yet another mode of applying the principle of reciprocity, 
which must be carefully distinguished from any previously stated in these 
pages. 

I have said that in place of the quantitative symbols of one alphabet, as 

X, y, z, &c., we may always substitute the operation symbols ~tz> ~s~ > 'Ti.y "^c. 

of the opposite alphabet. But now I say, in place of the quantitative symbols 

X, y, z, &c. occurring in the concomitant to any form /, may be substituted 

, , , . . ^ dF dF 

the quantities (observe, no longer operative symbols but quantities) -j^ , -j- , 

JET 

-jy , &c., F being itself any concomitant to /. Thus, for instance, taking F 
dQ 

identical with /, we see that f(j^, j- , ji., &c. j is concomitant to /: 
or again, if / be a function of x, y only, say f{x, y), taking F the polar 
reciprocal of /, that is /(-i?, ^), we see that/f--^, -^j'will be a 



342 On the Principles of the Calculus of Forms. [43 

concomitant to f: this concomitant, by the way it may be observed, will 

always contain / as a factor, because when f=0, x -^ + y-f^ = 0. Possibly 

it ma)'' be true that, when /" is a function of any number of variables 
X, y, z, &c., and F {^, tj, f, &c.) its. polar reciprocal, 

. /dF(oo, y, z, &c.) d¥ {x, y, z, &c.) 
■^ V dx ' dy ' ' 

which is a concomitant to /, contains / as a factor ; but I have not had time 
to see how this is. It is rather singular that Mr Cayley and Professor 
Borchardt of Berlin have both independently made to me the observation 

that, when f(x, y) is taken a cubic function of x and y, fir- , -r^^j is 

equal to the product of/ by the first evectant of the discriminant of f. 
The general consideration of the consequences of this new and important 
application of the idea of reciprocity must be reserved for a future section. 



Section V. Applications and Extension of the Theory of the Plexus. 

If j) = aaf + 4:ha^y + 6cx"y^ + idxy^ + ey*, 

we can obtain, by operating catalectically with x', y' upon 



,i,,d\. ( , d , d 



[^dx + ydy)'!'' ['d^ + y'dy^ 



the two concomitants 



ax^ + 2bxy + cy^, ha? + 2cxy + dy'^ 
hx- + 2cxy + dy-, ca? -f 2dxy + ey'^ 

a, b, c 

b, c, d 

c, d, e 



(1) 
(2) 



the one in fact being the Hessian, the other the catalecticant of itself. 
Again, if 

^ = aa^ + obx*y + lOcafy'^ + ... +fy^, 

by operating catalectically with x, y' upon the second and fourth emanants, 
as in the last case, we obtain the two covariants 



aa? + Sbx^y + Sexy'' + df, bafi + Scafy + Sdxy'' + ey'' I 
ba^ + iicx'y + Sdxy^ + ef, ca? + Ma?y + 'iexy'' -\- ff '' 



(1) 



43] 



On the Principles of the Calculus of Forms. 343 



(2) 



ax + hy, hx + cy, ex + dy 

hx + cy, ex + dy, dx + ey 

ex + dy, dx + ey, ex +fy 

which are in fact the Hessian and canonizant respectively of ^. So in 
general, for a function of x, y of the degree 2t or 2(. + 1, we can obtain i 
covariantive forms, the first being the Hessian, and the last the catalecticant 
on the first supposition and the canonizant on the second : calling the index 
of the function for either case n, the forms appearing in this scale will be 
of the degree (r + 1) in the constants, and of the degree (r + l)(m — 2r) in 
X and y. 

It has previously* been intimated that all these determinants admit of a 
remarkable transformation. 

This transformation may be expressed more elegantly by dealing not 
directly with the covariant forms as above given, but with their polar recipro- 
cants obtained immediately by writing ^ for — y and i; for x. 

(1) Suppose (^ = aa? + 2hx-y + 'S>exy- + dy^ ; 

a, 2b, c 

b, 2c, d 
r, 2^,7, V' 

will be found to be the reciprocant of its Hessian. 

(2) Let ^ = ax'+ibcfiy+ ... +ey*; 
the reciprocant of its Hessian will be found to be 

a, 36, 3c, d 

b, 3c, 3d, e 

(3) Let ^ = aaf + obafy + ... +fy^ ; 
the reciprocant of its Hessian will be 

a, 4<b, 6c, 

b, 

1^ 



4c, 

r. 



6d, 



^d. 


e 


4e, 


/ 


'?^ 




2^^, 


v' 



' [* p. 325 above, note +]. 



344 On the Principles of the Calculus of Forms. 



[43 



and the reciprocant of its canonizant is 



a, 


36, 


3c, 


d 


h, 


3c, 


3d, 


e 


c, 


U, 


3e, 


f 



The numerical coefficients in this and in the first case are inserted for 
the sake of uniformity, but it will of course be readily observed that when 
there is but one line of ^ and tj, that the numerical coefficients being 
the same, for each column may be rejected without affecting the form of 
the result. 



So again, if 














<!> = , 


za;" + Qboj^y + . . 


■ +9y\ 




the reciprocant of the H 


sssian 


is 








a, 


56, 


10c, Wd, 


5e, 


f 




h, 


be, 


lOd, lOe, 


■f. 


9 




r> 






■n\ 










^^ 


^H 


yf 



and the reciprocant of the second form in the scale, which comes between 
the Hessian and the catalecticant, is 



a. 


h, 'c, 


d, 


e 


6, 


c, d, 


e. 


f 


c, 


d, e, 


/. 


9 


r, 


^'v, h\ 


V', 






^^ r-v, 


Sv"; 


v' 



and so in general. The rule of formation is sufficiently plain not to need 
formulating in general terms. It is easy to see that all these forms are con- 
comitants to the function from which they are formed ; for example, take 



then 



form a plexus. 



(f) = a*" + 66a^2/ + • • ■ + 9y^ 

f—Xs ——(h (— 
\dx) ^' dx dy "' \dy. 



43] 



On the Principles of the Calculus of Forms. 345 



So likewise if we take -\|r = {x^ + yq)*, 

d^ ' dri 

form a plexus. But i/r and cj) are concomitantive, yjr being a universal con- 
comitant. Hence we may combine together these two plexuses, that is 

ax* + ^tbx^y + Qcx-y- + Mxy'^ + ey*\ 

bx* + 4!ca^y + Gdx'y- + iexy'^ + fy*y , 

ex* + 4<do6'y + 6ex-y' + 4fxy' + gyv 

par* -I- S^^Tjx^y + S^n^x^y^ + ■^'xy' 

^^x^y + 3p7?a;\v^ + S^t^^.*?/' + 97YJ 
and, by the principle of the plexus, x\ x^y. x'^y'^, xy^, y* may be eliminated 
dialytically, and the resultant will be the determinant last given, which is 
therefore a contravariant to (p. 

The manner in which I was led to notice this singular transformation is 
somewhat remarkable. 

In the supplemental part of my essay On Canonical Forms [p. 203 above], 
my method of solution of the problem of throwing the quintic function of 
two variables under the form u^ + v'^ + w^, led me to see that u, v, w are the 
three factors of 

ax + by, hx + cy, ex + dy 

bx + cy, ex + dy, dx + ey 

ex + dy, dx + ey, ex +fy 

the more simple mode of the solution of the same problem, given by me in 
the Philosophical Magazine for the month of November last [p. 266 above], 
led to 

d 



a, 


b, 


c, 


b, 


c, 


d. 


0, 


d, 


e, 


t, 


-xy\ 


x^y, 



■ a? 

as the product of the same three factors ; whence the identity of the two 
forms becomes manifest. In the paper last named I gave two proofs, one my 
own, the other Mr Cayley's, of a like kind of identity for the canonizant 
of any odd-degreed function of x, y in general. The proof of the identity 
of the corresponding forms in the much more general proposition above 
indicated [p. 325 above, footnote f] must be reserved until more pressing and 
important matters are disposed of. In the footnote referred to I ought to 
have added, in order to make the sense more clear, that the degree of the 
catalecticant there referred to in respect of the coefficients would be n. 



346 On the Principles of the Calculus of Forms. [43 

I regret to think that there are many other typographical errors in the 

earlier sections ; the most unfortunate of these is in tlie note at page [316], 

in the values of P and Q belonging to the cubic commutant dodecadic 

function of x and y, the corrected values of which will be given in my next 

communication. I ought also to observe, in correction of the remark made in 

the footnote to page [302], that it follows as a consequence of a recent paper 

by Dr Hesse iu Grelles Journal, that the method given by me in the text 

applied (according to what I have there termed the 1st process for obtaining 

an invariant resembling the resultant) to a system of three cubic equations 

d d d 
(in which application only the 1st powers of -j- , -j- , -y- enter) produces for 

that case also, as well as for the cases specified in the note, not a counterfeit 
resemblance of, but the actual resultant itself. 

Returning to the theory of the plexus of which I am about to enunciate 
a most important extension, I beg to refer my readers to the last paragraph, 
p. [291], in the last number of the Journal, where I have shown how to 
form, under certain conditions, a determinant by combining together various 
concomitants and eliminating dialytically one set of the variables, which 
determinant will be concomitantive to the concomitants out of which it is 
formed, and of course also therefore to their common original. 

Now the extension of this theorem, to which I wish to call attention, 
is this, that not only such determinant as a whole is a concomitant to such 
original, but every minor system of determinants that can be formed out of it 
will form a concomitantive plexus complete within itself to the same original. 
But, much more generally, it should be observed that there is no occasion 
to begin with a square determinant ; it is sufficient to have a rectangular 
array of terms formed by taking the several terms of one plexus or of several 
plexuses combined, provided that they are of the same degree in respect 
to the variables (or to the selected system of variables, if there be several 
systems), and forming out of such rectangular array any minor system of 
determinants at will. Every such system will be a concomitantive plexus. 
The simple illustrations which follow will make my meaning clear. 

Suppose 

^ = aa? + %ha?y + \bcafy'''+ 2\da?y^+ \bex'^y*+ Qfxy^ + gy^. 

I have previously remarked, in the foregoing sections, that a, b, c, d, e, f, g, 
the coefficients form an invariantive plexus to </> ; so also we know that the 
catalecticant 



h. 


0, 


d 


c. 


d, 


e 


d. 


e, 


f 


e, 


f, 


9 



43] 



On the Principles of the Calculus of Forms. 347 



is an invariant to 0. But we are now able to couple together these facts 
and see the law which is contained between them ; for if we take 

I being any number, as for instance, if we take t = 3, we shall have as a plexus 

aa? + Sbw'y + Sexy'' + dy\ 

ha? + Scx^y + Sdxy^ + ey^, 

CO? + Ma?y + 'iexy'' +fy\ 

da? + ^ex''y + ^fxy"- + gy^ ; 
accordingly not only is the determinant 

a, b, c, d 

b, c, d, e 

c, d, e, f 

d, e, f, g 

an invariant, but also the system obtained by striking out any one line and 
one column, being what 1 term the first minors, will be an invariantive 
plexus, so too will the system of second minors 

ac — b'', bd — c', ce — d', ad -be, ae — hd, be-cd, &c. 
form an invariantive plexus, as well as the last minors, that is, the simple 
terms a, b, c, d, e, f, g. Again, we might have taken the plexus 



which would give the array 



d\ 

dx) 


4>> 


d 
dx 


d 
dy 


<!>> 


\dy) 


^y 


a, 


b, 


0, 


d, 


e 




b, 


c, 


d, 


e, 


f 




c, 


d, 


e, 


/> 


9', 



but the minor systems of determinants herein comprised will be found to be 
identical with those last considered, with the exception that the highest 
system, containing a single determinant only, will now be wanting. So in 
general it will easily be seen that a similar method in general, when cj) is 
of 2t dimensions, will lead to t + 1 invariantive plexuses comprising the 
given coefficients grouped together at one extremity of the scale, and the 
catalecticant alone at the other ; and if (f) is of 2t + 1 dimensions, there will 
still be i + 1 such plexuses, commencing with the coefficients as one group 
and ending with a system of combinations of the (i + l)th degree in regard 
to the coefficients, which system accordingly takes the place of the cata- 
lecticant of the former case, which for this case is non-existent. 



348 On the Principles of the Calculus of Forms. [43 

As a profitable example of the application of this law of synthesis, in 
its present extended form, let it be required to determine the conditions that 
a function of oc, y of the fifth degree may have three equal roots. In general, 
let (f} = ax^+bhafy+lQca?y^+10dx^y^ + 5eccy*+fy^, then has a quadratic 
and cubic covariant of which I have written at large in my supplemental 
essay above referred to, being in fact the s and t (that is the quadrinvariant 
and cubin variant) in respect to x, y {x, y being treated as constants) of 

( , d , d 
\ dx " dy. 

Let these co variants respectively be called 

Ax^-\-2Bxy + Gy^ = u, 

(xa? + 3/3*^2/ + 'i'y^y^ + %^ = V ; 

Ax + By) 
then „ _, }■ 

Bx + Cy] 

forms a plexus, and 



will form another. 



^x^ + ^jxy + hf 



Now when a = 0, 6 = 0, c = 0, ^ will have three equal roots, and 

djX ^ dy) '^ 
becomes 

Qdy . x'Y' + 4 (dx + ey) x'y'^ + {ex +fy) y"^, 

of which the quadrinvariant in respect to «', y is easily seen to be dy 
and the cubinvariant d^y^. Accordingly the grouping 

A, B , 0, 

B, G ^"^^'""^ 0, d^ 



and the grouping 

a, 

/8, 7, S "— "'--= 0, 0, d 



a, /3, 7 , 0, 0, 

becomes 



\ A B \ 
Accordingly, we see that the determinant „' ^ and all the first minors of 

I i>, G I 
tt, B, 7 I 
I3 (, , that is a.'y — yS^, ^S — 7^, aS — ^y, become zero ; but the former 

single quantity d /^ being an invariant, and this last system being 

an invariantive plexus, all the quantities so affirmed to be zero will remain 
zero, notwithstanding any linear transformations to which (/> may be 
subjected ; thus then we obtain an immediate proof of the theorem that 



43] On the Principles of the Calculus of Forms. 349 

when a function of x and y of the fifth degree contains three equal roots 
the determinant of its quadratic covariant, which in fact is its sole quart- 
invariant, and the first minors of its cubinvariant will be all separately zero. 
This theorem may be made still more stringent ; for by combining 

Ax" + 2Bxy + Gy\ 

w)? + l^xy + '^y"-, 

yS«= + Ir^xy + S/, 

it becomes manifest that in the case supposed all the first minor deter- 
minants of 

A, B, G 

a. /3, 7 
/8, 7. « 

will be zero, showing in addition to the theorem last enunciated that also 
A : B : C :: a : 13 : J :: 13 : y: B. 

It is curious and instructive to remark that this last set of equations, 
stringent as they appear, and far more than enough to express a duplex 
condition, are not sufiicient to imply unequivocally the existence of three 
equal roots, unless we have also AG — B' = ; for suppose (p to take the form 
ax^ +fy^{b, c, d, e all vanishing); then it will easily be seen that 

8 = 0, 



a=0, /3 = 0, 7 = 0, 
^ = 0, B = af, G- 



0.* 



A, B, G 


and N= 


", P, y 




P, 7- S 





* If we take L, M, N a, system of fundamental invariants to 0, of which all the other 
invariants of <p are rational integer functions, then L = \A, B\ and the simplest forms for 31 and 
Ware \b, c\ 

M= A, B, G and N= a, 2^, y 

«, 2^, 7 
|8, 27, 5, 
|8, 27, 5 

where i and N are the discriminants of the quadratic and cubic covariants of <j> respectively, and 
a linear function of M, 1? is the discriminant of <p itself (L, M, N being of 4, 8, and 12 dimen- 
sions respectively in the coefficients of <^). 

For many purposes of the calculus of forms it is desirable to have the command of cases for 
which any two out of these three invariants may be made to vanish without the third vanishing ; 
and it will be found that when (p is of the form y^ {cx'+fif'), L = 0, M=0 ; when is of the form 
y (6a?'+/t/*), N=zO, i = 0; and when (p is of the form ax^ + ei/^, M = 0, N=0 ; and of course when 
<p is of the form y^ {dx^+fy^), L = 0, M = 0, N=0; it being obviously true in general, as remarked 
by Mr Cayley, that when not less than half the roots of a function of two variables are equal, 
all its invariants must vanish together. 



350 On the Principles of the Calculus of Forms. [43 

Consequently we shall still have all the first minors of 

A, B, C 

«, /3, 7 

/3, 7. 8 

zero, although there is not even so much as a pair of equal roots in ; AG — B^ 
however, it will be observed, is not zero in this supposition. 

The theory of Hessians, simple or bordered, may be regarded as one 
among the infinite diversity of applications of the principle of the plexus. 
Let U, V, W, &c. be any number of concomitants having the common system 
of variables x,y ... z. Let ;;^ represent 

, d , d , d 

da; dy '." dz' 
and take 

X^C/'+ XxF+ &c. + ;u,xTf = S" ; 

then dS dS dS 

dx' ' dy' '" dz' 

forms a plexus; and this, combined with x^> ^'^■■■■X^' enables us to 
eliminate dialy tically x', y', z, X ... /j,. The result is a Hessian of U, 
bordered with 

dV dV dV 

dx' dy " dz 

horizontally and vertically, and also with 

dW dW dW 
dx ' dy '" dz ' 

&c. &c. 

similarly dispersed ; which Hessian, so bordered, is thus seen to be a 
concomitant to U, V ... W. The Hessian, as ordinarily bordered with 
^, rj ... ^, is derived by taking for V the universal concomitant 

x^ + yr]+ ... +2^, 
and for W (if there be a double border) 

^S' + yv' + •■■ + z^', ' 

and so forth. 

If V be taken identical with U, the resulting form, consisting of U 

, , , .^. dU dU dU , , ,. * . 

bordered with -t— , -j-...-—, has been shown* in my paper "On certain 

general Properties of Homogeneous Functions," in this Journal, to be equal 

to the product of the simple Hessian of JJ and of U itself multiplied by a 

[* p. 173 above.] 



43] 



On the Principles of the Calculus of Forms. 351 



numerical factor. The theory of the bordered Hessian may be profitably 
extended by taking 

S = x''-U + \x^V+ ... +//-x'-TF, 

and combining with x^^ ■■•X^^ *^^ plexus obtained by operating upon 

8 with the rth powers and products of -5- , -7- . . . -r- , and eliminating 

dialytically the rth powers and products of cc', y ...z'. Thus if 

JJ = aa^ H- %a?y + Qcx'y'^ + 4>dxy^ + ey* and V= (x^ + yrjy, 

we obtain, by taking S=x*U+Xx'^' ^^^ proceeding as indicated in the 
preceding, 







a, h, 


c, r 








b, c, 


d, ^n 








c, d, 


e, v' 








P, H 


v% 




as a concomitant to U. 


So again, if 






[/■ = aa;" + 5ba:f*y + ... +fy^, 


we find 











ax + by, bx + cy, ex + dy, ^' 
bx + cy, ex + dy, dx + ey, ^tj 
ex + dy, dx + ey, ex +fy, if 



a concomitant to U. 



These extensions of the ordinary theory of Hessians will be found to 
be of considerable practical importance in the treatment of forms, for which 
reason they are here introduced. 



Section VI. On the Partial Differential Equations to Goncomitants, 
Orthogonal and Plagiogonal Invariants, c&c. 

In the 7th note of the Appendix to the three preceding sections* I alluded 
to the partial differential equations by which every invariant may be defined. 

This method may also be extended to concomitants generally. M. Aron- 
hold, as I collect from private information, was the first to think of the 
application of this method to the subject ; but it was Mr Cayley who com- 
municated to me the equations which define the invariants of functions of 
[* p. 326 above.] 



352 On the Principles of the Calculus of Forms. [43 

two variables*. The method by which I obtain these equations and prove 
their sufficiency is my own, but I believe has been adopted by Mr Cayley in 
a memoir about to appear in Grelle's Journal. I have also recently been 
informed of a paper about to appear in Liouville's Journal from the pen of 
M. Eisenstein, where it appears the same idea and mode of treatment have 
been made use of Mr Cayley 's communication to me was made in the 
early part of December last, and my method (the result of a remark made 
long before) of obtaining these and the more general equations, and of 
demonstrating their sufficiency, imparted a few weeks subsequently — 
I believe between January and February of the present year. 

The method which I employ, in fact, springs from the very conception of 
what an invariant means, and does but throw this conception into a concise 
analytical form. 

Suppose, to fix the ideas, 

^ = ax^ + ribx^"'^!/ + ^n{n—l) cx^^^y- + . . . + ly"^, 
and let I {a, b, c ...I) be any invariant to cf). 

Now suppose X to become x + ey, but y to remain unchanged ; the 

I 1 e t 
modulus of the transformation, ' , being unity, / cannot alter in con- 
sequence of this substitution ; but the effect of this substitution is to convert 
if> into the form 

ax^ + n^x'^~'^y ■\-\'n {n — 1) jx^'^y'^ + . . . + Xy"; 
■ where a = a, ^ = b + ae, y = c + 2be + ae^, &c. &c. 

\=l+ ... +n6e"-i + ae". 
■Consequently, if we make 

A6 = ae, Ac = 2be + ae^, &c. &c., 
we have by Taylor's theorem, observing that Aa = 0, 

A&-^,+&c.Y/+&c. = 0; 



' 1.2.3\ db 

* It is extremely desirable to know whether M. Aronhold's equations are the same in form 
as those here subjoined. It is difficult to imagine what else they can be in substance. Should 
these pages meet the eye of that distinguished mathematician he will confer a great obligation 
on the author and be rendering a service to the theory by communicating with him on the 
subject: and I take this opportunity of adding that I shall feel grateful for the communication 
of any ideas or suggestions relating to this new Calculus from any quarter and in any of the 
ordinary mediums of language — French, Italian, Latin or German, provided that it be in the 
iatin character. 



43] On the Principles of the Calculus of Forms. 353 

and this being true for all the values of e, every separate coefficient of e 
in A7 must be zero : hence we obtain n different equations by equating 
to zero the coefficients of e, e' ... e" respectively. The first of these equations 
will be 

and it is obvious that this will imply all the rest; for, when e is taken 
indefinitely small, I {a, b, c ...) does not alter (when this equation is satisfied) 
by changing a, b, c ... into a', b', c'...; consequently I (a', V, c', &c.) 
will not alter, when in place of a , b', c we write a", b" , c" , &c., obtained 
from a', b', c, &c., by the same law as a, b', c', &c., from a, b, c, &c. 

Thus we may go on giving an indefinite number of increments, ey to x, 
without changing the value of /. Consequently, if the equation above 
written be satisfied, a priori all the rest must be so too. But there is not 
any difficulty in showing the same thing by a direct method*. 



For we have 



a4 + 26 J- + 3c :^ + fee.) / = 0, 
db do dd / 



an identical equation. Hence 



hence 



that is 



a-J + 26 4 + 3c-^, + &c.Va4 + 26^ + 30-^, + &c. 



' db dc dd ' )\ db " do dd 



+ ^V6 + 26| + 3o^ + &c.K = 0, 



repeating the application of the symbolic operator 
.| + 26| + &c.), 

* The method above given has the advantage however of being immediately applicable to 
every species of concomitant, and we learn from it that concomitance, whether absolute or 
conditional, is sufficiently determined when affirmed to exist for infinitesimal variations ; it 
cannot exist for infinitesimal variations without, by necessary implication, existing for finite 
variations also ; a most important consideration this in conducing to a true idea of the nature 
of invariauce and the other kinds of concomitance, and in cutting off all superfluous matter from 
the statement of the conditions by which they are defined. 

s. 23 



354 On the Principles of the Calculus of Forms. [43 

we obtain 

1 . 2 . 3 L^ + 46 ^^ + 10c J^ + &c. 

and so on ; the numerical multipliers of the terms of the several series 
within the parentheses forming the regular succession of figurate numbers 

1, 2, 3, &c. 

1, 3, 6, &c. 

1, 4, 10, &c. 

It is easy to see that these equations correspond to the results of making 
the coefficients of the successive powers of e equal to zero. 

I may remark, that the first instance as far as I know on record of this, 
(as some may regard it rather bold) but in point of fact perfectly safe and 
legitimate method of differentiating conjointly operator and operand, occurs 
in a paper by myself in this Journal, Feb. 1851, " On certain General 
Properties of Homogeneous Functions" [p. 165 above] ; where I have applied 
it in operating with 



upon 






which, as I have there noticed, gives the result 

d )''■ 

The equation f a ^^ + 26 t- + &c.) /= is evidently not enough to define 

/ as an invariant; it merely serves to show that / does not alter when 
in place of a; we write x + ey, but this is true for any function of the 
differences of the roots of the form multiplied by a suitable power of a, 
namely that power which is just sufficient to cause the product to become 
integer. But if we now, for convenience, write 

j> = ax^ + nhx^-'^y + \n{n— 1) cx^-^y^ + . . . 

■\-\n{n—\) c'x-y'^"- + nh'xy^'"^ + a'y", 



43] On the Principles of the Calculus of Forms. 355 

and form the similar equation from the other side, namely 
d ^1, d „ , d 



{^■i^^^i^^a^"')''"- 



these two equations together will suffice to define any invariant, as I shall 
proceed to show — these are the two equations alluded to brought under 
my notice by Mr Cayley. If they coexist, it follows from the method by 
•which I have deduced them that x may be changed into x + ey, or y into 
y +f!K, without / being altered, e and / having any values whatever : and 
it is obvious that these substitutions may be performed, not merely alter- 
natively but successively, because the equations between the coefficients 
are identical equations, and depend only on the form of /. 

Let now x become x + ey, and then y become y +fx; the result of these 
substitutions is to convert 

X into X + efx + ey, 
and y into fx-\-y. 

Finally, let x become x + gy; then x is converted into (1 + e/) (a; + gy) -t- ey, 
and y into y +f{x + gy), 

that is X becomes (1 + ef) x + (eg + efg) y, 

and y becomes fx + (1 -\-fg) y. 

The modulus of substitution it is evident, a priori, always remains unity, 
and nothing would be gained by pushing the substitutions any further, as it 
is clear that we may satisfy the equations 

\ + ef=p, e + g + efg = q, 

y=p', i+fg = q', 

for all values of p, q, p', q, which satisfy the equation 

pq'-p'q = l, 
and for none other except such values ; hence / remains unaltered for any 
unit-modular linear transformation of x, y, and is therefore an invariant 
by definition. 

If <j) be taken a function of three variables, x, y, z, and be thrown under 
the form 

a2" + iflix + h{y) z^-^ -|- {a^'' + %h^y -f Cat/') 2"-' + &c., 
and I be any invariant of </>, by supposing x to become x -f ey, and giving 
\, 62, C2, &c., the corresponding variations, and taking e indefinitely small, 
we obtain 

' + (-.i + ^'-D + (»■ * * ^"'k + '"-iy +M '-"■ 



'db. 



23—2 



356 On the Principles of the Calculus of Forms. [43 

and in like manner, by arranging ^ according to the powers of y and of x, we 
obtain two other pairs of equations : it is clear, however, that three equations 
(it would seem any three out of the six) would suffice and imply the other 
three. The method of demonstration will be the same as in the instance of 
two variables : First, it can be shown by the method of successive accretions, 
that / remaining invariable when x receives an indefinitely small increment 
ey, or y an indefinitely small increment ez, or z an indefinitely small increment 
e*', it will also remain invariable when these increments are taken of any 
finite magnitude. Secondly, by eight successive transformations, admissible 
by virtue of the preceding conclusion, x, y, z may be changed into any linear 
functions of x, y, z, consistent with the modulus of transformation being unity. 
And in general for a function of m variables, m partial differential equations 
similarly constructed (but not however arbitrarily selected) will be necessary 
and sufficient to determine any invariant : and it is clear that all the general 
properties of invariants must be contained in and be capable of being educed 
out of such equations. 

The same method enables us also to establish the partial differential 
equations for any covariant, or indeed any concomitant whatever. 

Thus let 

^ = aa;" + nbx^~^y + \n{n-\) cx^'Y' + • • ■ + nh'xy'^'^ + o!y^ = 0, 
and let K{a, b, c, &c. ; x, y, x', y', &c. ; f, r), &c.) represent any concomitant, 
X, y ; x', y being cogredient, and ^, r\, &c. contragredient systems ; when 
X, y become x + ey, y, any such system x', y' becomes x + ey', y' ; and any 
such s^'stem as ^, ?? becomes ^,ri — e^; and taking e indefinitely small, the 
second coefficients a, b, c, &c. become a, b + ae, c + 2be, &c. as before ; hence 
the equation to the concomitant becomes 

d -,, d d , d i. d r. ] n^ 

'db+'^dc + --yrx-yd^'+- + ^rn-^'T'*' 

and in like manner, by changing y into y + ex, results the corresponding 

equation 

, d ^,, d d , d d o ] T;r n 

a-jT, + 2b ^-r + ... — X-, X T— ,+ ...+n-rr. — &c.^ii=0. 

do dc dy dy d^ J 

These two equations define in a perfectly general manner every concomitant 
(with any given number of cogredient and contragredient systems) to the 
form <f> ; and the due number of pairs of similarly constituted equations 
will serve to define the concomitant to a function of any given number 
of variablesf. 

* For we have 

A' (a, b + ae, c + 2be, &c.; x, y, &c. ; |, rj, &c.) 
= K{a, b, c, &c.; x, x + ey, &c.; |, n-e^, &o.; &e.). 
+ Vide Note (10) [p. 361 below]. 



43] On the Pi-inciples of the Calculus of Forms. 357 

In like manner we may proceed to form the equations corresponding 
to what may be termed conditional concomitants, whether orthogonal or 
plagiogonal. The concomitants previously considered may be termed absolute, 
the linear transformations admissible being independent of any but the one 
general relation, imposed merely for the purpose of convenience, namely of 
their modulus being made unity. An orthogonal concomitant is a form 
which remains invariable, not for arbitrary unit-modular, but for orthogonal 
transformation, that is for linear substitutions of «, y ...z, which leave 
unchanged x- + y- + ... + z-: in like manner, a plagiogonal concomitant may 
be defined of a form which remains invariable for all linear substitutions 
of X, y ... z, which leave unaltered any given quadratic function of x,y ... z. 
Thus, let it be required to express the condition oi Q{a,h, c ... x, y; ^, i?), 
being an orthogonal concomitant to the form 

ax^ + nbx^~^y + . . . + nb'xy'"'~'^ + a'y". 

Let X become x + ey, e being indefinitely small, then y must become y — ex, 
and the variations of a,h ... V, a' will be the sum of the variations produced 
by taking separately x + ey for x and y — ex for y. Hence the one sole 
condition for Q being of the required form becomes 



d ^j d d ^ ^ ^ 

db+'-''do + --ydx + ^dv) 
, d „,, d d 



• = 0, 



dc' '" dy d^J 

or, as it may be written, 6Q — ccQ = 0, where 0Q = 0, a)Q = are the two 
equations expressing the conditions of Q, being an unconditional or absolute 
concomitant ; and so in general if (^ be a function of m variables, we may 
obtain |m(m-l) equations of the form L-M=0 for the concomitant, 
of which however (m— 1) only will be independent. 

Supposing, again, the substitutions to which x, y are subject to be 
conditioned by Ix^ + 2mxy + ny^ remaining unalterable, or which is a more 
convenient and only in appearance less general supposition by x'' + 2mxy + y"^ 
remaining unalterable, the general type of an infinitesimal system of substi- 
tutions will be rendered by supposing x, y to become (1 -I- me) x + ey, 
— ex + O-— me) y, respectively, for then x^ + 2mxy + y"^ becomes 

(1 - mV) x^ -f {2m -h (2m - 2?w^) e-\ xy -^(^ — m-e^) y'^, 

which differs from x'^ + 2mxy + y^ only by quantities of the second order 
of smallness which may be neglected, and ^ and r] will therefore become 
{1 —me) ^— ei], —ex + {l + me)y, respectively: then, as to the coefficients 
of <p, in addition to the variations which they undergo when m is zero, there 
will be the variations consequent upon x assuming the increment Tnex, and y 



358 On the Principles of the Calculus of Forms. [43 

the increment — mey : but by making x become x + mex, a, b, c, &c., h', a' 
assume respectively the variations 

n . mea, (n — l) meb, . . . meb', 0, respectively ; 

and by making y become y — mey, the corresponding variations become 

0, — meb, ... — (n — 1) meb', — n . mea', respectively. 

Hence the equation becomes 

where 6 and « have the same signification as before, and where \ denotes 



na-+(n-l)b^^+...+b _ + .^-^-, 



and fi denotes 



d ^ d , d d d 

db dc da' " dy drj' 



If there be several systems of x, y or of ^, rj, or of both, the only difference 
in the equation of condition will consist in putting 

^{"i)' ^(^|)' ^("s)- ^{"i)' 

H4y H^i)' H4)' H^i 

instead of the single quantities included within the sign of definite sum- 
mation. 

Fearing to encroach too much on the limited space of the Journal, 
I must conclude for the present with showing how to integi-ate the general 
equation to the orthogonal invariant of (p, the general function of x, y. 

Beginning with <^ = ax" + 2bxy + cy"^, the equation becomes 



Write now 



we have then 



da = — 2bdd, dx = ydO, 
db = (a — c) dd, dy = — xdd, 
dc = + 2bde ; 



\da + fidb + vdc = dd {fia + 2(v —X)b— /u.c}. 
Let fi = k\, 2{v — X) = Kfi, — fi = Kv; 

then d log (Xa + /j,b+ vc) = Kdd ; 

or \a + fib+ vc = be"^. 



43] On the Principles of the Calculus of Forms. 359 

To find K we have the determinant 

a;, -1, I 

2, «:, -2 1 = 0, 

0, 1, k\ 
that is, «= H- 4k = 0, 

and calling the three roots of this equation «i, k^, k^, we have 

Ki ^ Uj K^ ^ ^ij ^^3 ^ ^t J 

accordingly we may put 

K = 0, \=1, fl = 0, v=l, 

or K = 2t, X = 1, /x = 2t, V = — 1, 

or « = — 2t, X = l, /i = — 2t, !/ = — 1. 

Again, pdx + qdy = {py — qx) dO ; 

and putting —q = ep, p = eq, so that px +qy = Ee^^, 

and we may put 

e = I, ^ = 1, q = — i, 

or e — — i,p = l,q= + i. 

Consequently the complete integral of the given partial differential equation 
is found by writing 

a + c = l, x—iy = Ee!-^, 

a + 2t6 - c = Z'e='^ x -\- iy = E'e^'^, 

a — 2ib — c = Z"e~-'^. 

By means of these iive equations, after eliminating 6, we may obtain four 
independent equations between a,b,c; x, y. Suppose 

Qi = 0, Q, = 0, Q3 = 0, Q4 = 0; 

then Q = F{Q^, Q^, Q^, Q^) is the complete integral required. 

Pursuing precisely the same method for the general case, it will be found 
that, calling the degree of the given function n when n is even, the equation 
in K to be solved will be 

K (k' + 4>) (k- + 9) ... (/c' + n^) = ; 
and when n is odd (say 2m + 1), the equation in k to solve will be 
(« + !)(/<:'+ 9)... (K^+n2) = 0; 



360 On the Principles of the Calculus of Forms. [43 

and performing the necessary reductions, and calling the roots of the 
equation, arranged in order of magnitude, k^l, «„t ... A;„t, respectively, it will 
be found that the equations containing the integral become 

X— cy — Jie'' 



Ln+-L = ?«+i e"»+'" 



where li, In ... In+ii E, E' are arbitrary constants, and where L^, L^... Ln+i 
are the values assumed by the 1st, 2nd ... (n + l)th coefificients of the given 
function <p, or 

aa;" + nhx^'~'^y + . . . + nh'xy^^'^ + a'y''\ 
when it is transformed by writing x + iy in place of x, and y + ix in place oiy. 
L is of course employed in the foregoing according to the usual notation to 
represent ^J{— 1). The same method applies to the general theory of plagio- 
gonal concomitants, where the linear substitutions are supposed such as to 
leave Ix'' + Imxy + ny"- unaltered in form, and the equations in 6 which 
contain the integral present themselves under a similar aspect. But a more 
full discussion of these interesting integrals must be reserved until the 
ensuing number of the Journal. 

NOTES IN APPENDIX. 

(9) The scale of covariants to a function of {x, y) obtained by the 
method of unravelment [on p. 297 above], may be otherwise deduced 
in a form more closely analogous to that of the corresponding theorems 
for the corresponding invariantive scale [on p. 295 above], by a method 
which has the advantage of exhibiting the scale equally well for the case 
of functions of the degree 4t + 2 or 4t + 4, the only difference being that 
in the latter case the coefficients of the odd powers of X will be found all 
to vanish, so that the degrees of the covariants will rise by steps of 4 instead 
of by steps of 2, just conversely to what happens in the invariantive scale ; 
whereas in the invariantive scale alluded to the forms containing odd powers 
of X vanish when the degree of the function is of the form 4t + 2, but do not 
vanish when it is of the form 4f. This method in the form here subjoined 
is a slight modification of one suggested to me by my friend Mr Cayley. 

Let F be the given function of x, y of the degree 2n ; take the systems 
x', y' ; a?!, y-^ cogredient with one another and with x, y. Then form the 
concomitant 



43] 



On the Principles of the Calcuhis of Foj'ms. 



361 



Then (by what may be termed the Divellent method, which has been pre- 
viously applied by me in the Philosophical Magazine for Nov. 1851) 
calling 6oi ^i> 0^ •■■ On, the coefficients of 

■^'n-i „' _ y'n Jjj ^^ 



we shall have 



6, = A,x^ + B,x"-'y + . . . + X,2/», 

On = AnOd"- + Bnof-^y + ...+ L^y"^ , 

the coefficients being functions of the coefficients of f and of quadratic 
combinations of «!, 2/1, affected with the multiplier \ ; and the determinant 

A^, B^ ... La 

A„ B,...L, 



An,Bn...Ln 



will give a function of X in which the coefficients of the several powers of X 
will be all zero or covariants of F. 

The actual form of this determinant is not here given for want of space 
and time, but will be exhibited hereafter. Precisely an analogous method 
applies to obtain the scale to (x, y, zf given in Note (2) [p. 322 above]. 
Calling F={x, y, z)*, let the systems x', y , z; x-y,y-^,z^, be taken cogredient with 
one another and with x, y, z. Then, using R to express the determinant 

x', y', z' 

X, y, z 

«i, TJi, Zi 



t d , d , d\ „ ^ -n 

+ 2/^ + ^-1 ^+^^' 



and making 

~ V dx^ ^ dy ' " dzj 
and proceeding as above by the divellent method, we obtain the scale required. 

(10) [p. 356 above.] It is obvious that these defining equations ought 
to give the means of discovering and verifying all the properties of con- 
comitants ; but it is very difficult to see how in the present state of analysis 
many of the general theorems that have been stated, readily admit of being 
deduced from them. 

The comparatively simple but eminently important theory of the evector 
symbol does however admit of a very pretty verification by aid of these 
equations. Thus, suppose 6 any concomitant ; suppose a contravariant to a 
function F of x, y, say 

««" + nbx^'^y + ... + nb'xy"~^ + a'y"'. 



362 On the Principles of the Calculus of Forms. [43 

Then 6 must satisfy the two equations 

where i = a^r + 26 -7- + ... +n6'^^, , 

do etc da 



r, , d „., d , d 

i = a St/ + 2& -7-' + • • • + "^ T- ■ 
do dc da 



Now let ^ = ;\; {6) where 



^=^"^+^"-^4+^"-^''^l+-+''"^= 



then L{x0) = xiLd)-(xL)d 

Hence (z + f |) ^ W = x {(i + ?|) ^ = x(0) = 0. 

Similarly (i' + ^ A) ^ (^) = 0. 

Hence if 6 is an integral of the two conditioning equations, so also is ^ (^)- 
In like manner, if ^ be a co variant or any other kind of concomitant of F, 
it may be proved that its evectant % {6) is the same. 

(11) [p. 331 above.] Very much akin with the supposed equations is the 

following most remarkable equation, which can be proved to exist. Let 

be a function of x and y of the 5th degree. Let P and Q be the quadratic 

and cubic covariants of ^. P is of two dimensions in the coefficients and 

also in the variables, and Q of three dimensions in both; they are in fact 

[ , d d\* 

the s and t (in respect to x' and y') of (x -=- +y' -j-] cfi. Then, giving P and 

Q proper numerical factors, it will be found that 

ffn4> + PHcf) + Q(/) = 0. 

I believe that a similar equation connects any function of x and y above 
the 3rd degree with its first and second Hessians. The proof will be given 
in a subsequent Section, where also I shall give a complete proof, which 
occurred to me immediately after sending the preceding note to the press, 
of the complete Theory of the Respondent by means of the general equations 
of concomitance. 



43] On the Principles of the Calculus of Forms. 363 

P.S. Since the preceding was in type, I have ascertained the existence 
and sufficiency of a general method for forming the polar reciprocal and 
probably also the discriminant to functions of any degree of three variables 
by an explicit process of permutation and differentiation. In particular 
I am enabled to give the actual rule for constructing the polar reciprocal 
and the discriminant curves of the 4th and oth degrees. So far as regards 
the polar reciprocal of curves of the 4th degree M. Hesse has already given 
a method of obtaining it, but mine is entirely unlike to this, and rests upon 
certain extremely simple and universal principles of the calculus of forms. 
The only thing necessary to be done in order to carry on the process to 
curves of the 6th or higher degrees, is to ascertain the relation of the 
discriminants of functions of two variables of those respective degrees to such 
of the fundamental invariants as are of an inferior order to the discriminant. 

The theory applies equally well to surfaces and to ftmctions of any 
number of variables, and may, I believe, without any serious difficulty be 
extended so as to reduce to an explicit process the general problem of 
effecting the elimination between functions of any degree and of any number 
of variables. The method above adverted to will appear in a subsequent 
Section. 

[Continued pp. 402 and 411 belou'.J 



44. 



SUR UNE PROPRI^TE NOUVELLE DE L':^QUATION QUI 
SERT A DETERMINER LES INEGALITES SECULAIRES 
DES PLANETES. 

[Nouvelles Annates de MatMmatiques, xi. (1852), pp. 438 — 440.] 
[Extract.] 

6. Soit le determinant carre symetrique 



^. 1 J ^2, 2 ... ^2, n 



^71, 1 > ^71, 2 • . . ^n, n 

dans lequel on a, d'apres la definition, 

£levant le determinant a la puissance p, on obtient le determinant 

-4i,i, Ai^n ... Ai^n 
-^2,1» -^2,2 •■• ■"■2,71 



(M) 



An,,, A,, 



An,n 



(N) 



et ce determinant est symetrique aussi par rapport a, la diagonale Ai^x> 

A2-" A«- 

Retranchant de chaque terme de la diagonale symdtrique de (M) la meme 
quantite X, on ohtient le determinant 



0^1,1 — A., (Ii^2 ... (h,n 

^2, 1 J ^2, 2 '^ ... ^2, n 

^71, 1 J ^71, 2 • . • 0^71, 71 ^ 



(P) 



44] 



Sur une propriete nouvelle. 



365 



Developpant ce determinant et ordonnant par rapport d X, on obtient une 
expression qui, dtant egalee d zero, donne I' Equation 

X" -/X"-' + £rX»-^ + . . . (- 1 )" « = 0, (1) 

Equation qui a n racines r^elles (voir t. X. p. 259). 

Retranchant de chaque terme de la diagonale symetrique du determinant 
(N) la quantite yx, et operant comme ci-dessus, on parvient a, V equation 

^n _ ^^«-i + g.^„-2 + . . . (_ 1 )« r = 0, (2) 

equation qui a aussi n racines reelles. Les racines de cette Equation sent 
les racines de V equation (1), elevees chacune d la puissance p. 

Demonstration. Repr^sentons par 
les p racines de I'e'quation pP — l=0. Ecrivons le determinant 

1,1 Rq^' (^1,1 • ■ • ^i,n 
2, 1 ) *^2, 2 Rg*^ ' ' ■ ^2, n 



(^n, 1 ^n, n Rq^ 

et faisons q dgal successivement a tons les nombres de la suite 1,2, S ... p, 
on aura p determinants ; le produit de tous ces determinants reste evidem- 
ment le meme dans quelque ordre qu'on prenne ces determinants, et, d'apres 
les propridtes connues des racines de I'unit^, tous les termes en p qui ne 
seront pas Aleves a une puissance p disparaitront, et X accompagnant toujours 
p, il ne reste done que des X^*, et le ddterminant-produit sera 

-"' n ■■•' -^2 o— X* ... ^j.Jl . (Q\ 



■^n,i> -^n^i, -a.n,7j~^^ 

ou, faisant abstraction de X, on a le determinant (N). Ainsi 

/i = X^. 



C. Q. F. D. 



7. Application. 
determinant 



w = 2, et p = 2; 

\ a, b \ 
\b, c I ' 

elevant ce determinant au carrd, on a 

I a" + 6S ab + hc 
I ab + bc, b'^ + c^ 



(M) 



(N) 



366 Sur une propriete nouvelle. [44 

determinant , ' , , (V\ 

I 6, c — X I ^ ' 

"S? - {a + c)\ + ac -h" = ; (1) 

\ a? + P- H., ab + bc I 



determinant 



Faisons 



\ ab + bc, b^ + c'~/j,\' 

jj," - (a' + c^ + 260 /J' + (ac- ¥y = 0, oh fi = X=. (2) 



71 = 2, p=3, 

(K) ne change pas, et Ton a 

I a^ + 2ab^ + b'c, a% + abc + ¥ + bc"- 

a% + abc + b^ + b-c, a¥ + 2¥c + c' 



(N) 



le determinant (P) et Fequation (1) restent les memes; mais I'equation 
(2) devient 

fj} _ {a? + &+ Sab- + 3c¥) fi + (ac- bj = 0, * 
oil /i = \=, 

car, Xi et X, etant les deux racines de I'equation (1), on a 

\^ + V = a' + d' + Sab'' + 3c¥, Xi' X^' = (ac - by. 

8. M. Sylvester fait observer que son theoreme est un cas particulier 
d'un thdoreme plus general, demontre par M. Borchardt, pour des determi- 
nants quelconques, et qui devient le theoreme demontre ci-dessus, lorsque 
le determinant est sym^trique {Journal de MatJiematiques, t. xii. p. 63, 1847). 



45. 



ON A REMARKABLE THEOREM IN THE THEORY OF EQUAL 
ROOTS AND MULTIPLE POINTS. 

[Philosophical Magazine, iii. (1852), pp. 375 — 378.] 

In order that the theorem which I propose to state may be the more 
easily understood, and with the least ambiguity expressed, I shall commence 
with the case of a homogeneous function of two variables only, x and y. 

Let 

(j) = aos"' + nhx'"'~^y + \n{n — V) ox^~'^y" + . • . + nb'xy^~^ + a!y^, 

and let the result of operating with the symbol 

on any function of a, b, c ... b', a' be called the Evectant of such function, 
and the result of repeating this process r times the rth Evectant. 

Understand by the multiplicity of the equation the number of equalities 
between the roots that exist ; so that a pair of equal roots will signify a 
multiplicity 1, two pairs of equal roots, or three equal roots a multiplicity 2; 
a pair of equal roots and a set of three equal roots, a multiplicity 1 + 2 or 3, 
and so on. Now suppose the total multiplicity of ^ to be m : the first part 
of the proposition consists in the assertion that the 1st, 2nd, 3rd ...(m — l)th 
Evectants of the discriminant of (/>, that is of the result of eliminating x and 

V between -^ , ^ (as well as the discriminant itself), will all vanish in 
^ dx ay 

whatever way the multiplicity is distributed ; the second part of the 

proposition about to be stated requires that the mode should be taken 

into account of the manner in which the multiplicity (m) is made up. 

Suppose, then, that there are r groups of roots, for one of which the 



368 On a remarkable Theorem in the [45 

multiplicity is mj, for the second m^, &c., and for the rth m^, so that 
m,,+ m^ + . . . + 'mr= m. Then, I say, that the mth evectant of the deter- 
minant of <^ is of the form 

where a^-.h^, a^-.b^.-^ar: br are the ratios oi x : y corresponding to the several 
sets of equal roots. 

This latter part of the theorem for the case of m = 1 was discovered 
inductively by Mr Cayley, by considering the cases when ^ is a cubic, 
or a biquadratic function. I extended the theory to functions of any 
number of variables, and supplied a demonstration, that is for the case 
of one pair of equal roots. Mr Salmon showed that my demonstration could 
be applied to the case of two pairs of equal roots, or two double points, 
&c., and very nearly at the same time I made the like extension to the case 
of three equal roots, cusps, &c., and almost immediately after I obtained 
a demonstration for the theorem in its most general form. This demon- 
stration reposes upon a very refined principle, which I had previously 
discovered but have not yet published, in the Theory of Elimination. 

I have here anticipated a little in speaking of the theorem as applicable 
to curves and other loci. 

Suppose ^ {x, y, a) = to be the equation to a curve expressed homo- 
geneously. 

Let 

^ (x, y, z) = ax"' + (na'x^-^y + nb'x"~^z) 

+ ^w (n - 1) a"«"-y + n(7i- 1) U'x'^-^yz + Jn (n - 1) c"a;»-^i;=, 

+ &c. &c., 

and understand by the evectant of any quantity the result of operating upoa 
it with the symbol 

da ^ da db " da 

Suppose, now, the curve to have double points, the {r — l)th evectant 

(and of course all the inferior evectants) of the discriminant of ^ (meaning 

,. . . , , ddi dd) d<h\ .,, 

thereby the result of elimmatmg x, y, z between -^, -^, -r- j will 

all vanish, and the rth evectant will be of the form 

{a^x + b^y + c^zy x {a^ + % + '^i^Y • • • ^ («r« + bry + c,.zY, 

where ai:6i:Ci, asibtiCz ...aribr'.Cr are the ratios of the coordinates at 
the respective double points. If there be cusps the multiplicity of each 



45] Theory of Eqtial Roots mid Multiple Points. 369 

such will be 2 ; and calling the total multiplicity m, to every cusp will 
correspond a factor of the 2?)th power in the with evectant ; and so on in 
general for various degrees of multiplicity at the singular points respectively. 
The like theorem extends to conical and other singular points of surfaces ; 
so that there exists a method, when a locus is given having any degree of 
multiplicity, of at once detecting the amount and distribution of this multi- 
plicity, and the positions of the one or more singular points. In conclusion 
I may state, that precisely analogous results {mutatis mutandis) obtain, 
when, in place of a single function having multiplicity, we take the more 
general supposition of any number of homogeneous functions being subject 
to the condition of pluri-simultaneity, that is being capable of being made 
to vanish by each of several different systems of values for the ratios between 
the variables. Multiplicity in a single function is, in fact, nothing more 
nor less than pluri-simultaneity existing between the functions derived from 
it by differentiating with respect to each of the given variables successively. 
But as I purpose to give these theorems and their demonstration, which 
I have already imparted to my mathematical correspondents, in a paper 
destined for reading before the Royal Society, I need not further enlarge 
upon them on the present occasion. 

P.S. In the above statement I have spoken only of cusps of curves which 
are the precise and unambiguous analogues of three coincident points in 
point-systems, in order to avoid the necessity of entering into any disquisition 
as to the species of singularity in curves or other loci corresponding to 
higher degrees of multiplicity in point-systems, a subject which has not 
hitherto been completely made out. I may here also add a remark, which 
gives a still higher interest to the theory, which is (to confine ourselves, for 
the sake of brevity, to functions of two variables), that if any root of x:y, 
say a:h, occur \ + jjl times, the total multiplicity of the equation being 
supposed m, and its degree n, then taking t any integer number not exceed- 
ing fi, the (m -I- t)th evectant of the discriminant will contain the factor 
{ax+hy)^-'-'!'^. So that, for instance, if there be but a single group of equal 
roots, and they be 1 -f /x in number, every evectant up to the {fi — l)th 
inclusive will vanish, and from the jjXh to the (2/i — i)th will contain a power 
of {ax 4- hyY. 



24 



46. 

OBSERVATIONS ON A NEW THEORY OF MULTIPLICITY. 

{Philosophical Magazine, III. (1852), pp. 460 — 467.] 

In the Postscript to my paper in the last number of the Magazine, 
I mis-stated, or to speak more correctly, I understated the law of Evection 
applicable to functions having any given amount of distributive multiplicity. 
The law may be stated more perfectly, and at the same time more concisely, 
as follows. Every point represented by the coordinates ai, /3i...7i, for 
which the multiplicity is m^, will give rise in every evectant* of the discrimi- 
nant of the function to a factor {a:^x + ^^y + ... +jiz)"^^"', n being supposed 
to be the degree of the function. Hence if there be r such points, for which 
the several multiplicities are wij, «i2...my, every evectant must contain 
(mj + m^+ ... + nir) n linear factors ; and as the ith evectant is of the degree 
m, it follows that all the evectants below the («ii +m2+ ... + mr)th evectant 
must vanish completely, and this Evectant itself be contained as a factor 
in all above itf. When a function of only two variables is in question, there 
is no dif3Sculty in understanding what property of the function it is which 
is indicated by the allegation of the existence of multiplicities nv^, ni„ ... m^; 

* Frequent use being made in what follows of the word Evectant, I repeat that the evectant 
of any expression connected with the coefficients of a given function (supposed to be expressed 
in the more usual manner with letters for the coefficients affected with the proper binomial or 
polynomial numerical multiphers) means the result of operating upon such expressions with a 
symbol formed from the given function by suppressing all the binomial or polynomial numerical 
parts of the coefficients to be suppressed, and writing in place of the literal parts of the coeffi- 
cients a, b, c, &c. the symbols of differentiation -r- , tt > -r . &o. ; in all that follows it is the 

da do ac 

successive evectants of the discriminant alone which come under consideration. I need hardly 
repeat, that the discriminant of a function is the result of the process of elimination (clear from 
extraneous factors) performed between the partial differential quotients of the function in respect 
to the several variables which it contains, or to speak more accurately, is the characteristic of 
their coevaneseibility. 

t The constitution of the quotients obtained by dividing all the other evectants of the 
discriminant by the first non-evanescent one, presents many remarkable features which remain 
yet to be fully studied out, and promise a wide extension of the existing theory. 



46] On a new Theory of Multiplicity. 371 

as already remarked, this simply means that there are r distinct groups 
of equal roots, such groups containing l+wii, l+ma-.-l + TWr roots re- 
spectively. So for curves and higher loci, the total distributive multiplicity 
is the sum of the multiplicities at the several multiple points. But the true 
theory of the higher degrees of multiplicity separately considered at any 
point remains yet to be elaborated, and will be found to involve the considera- 
tion of the theory of elimination from a point of view under which it has 
never hitherto been contemplated. 

Confining our attention for the present to curves, we have a clear notion 
of the multiplicity 1 : this is what exists at an ordinary double point. As 
well known, its analytical character may be expressed by saying that the 
function of x, y, z, which characterizes the curve, is capable, when proper 
linear transformations are made, of being expanded under the form of a series 
descending according to the powers of z, such that the constant coefficient 
of the highest power of z, and the linear function of x, y, which is the 
coefficient of the next descending power of z, may both disappear. Again, 
when the multiplicity is 2, the third coefficient, which is a quadratic function 
of X and y, will become a perfect square. This is the case of a cusp, which, 
as I have said, is the precise analogue to that of three equal roots for a function 
of two variables. Before proceeding to consider what it is which constitutes 
a multiplicity 3 for a curve, it will be well to pause for a moment to fix the 
geometrical characters of the ordinary double point and the cusp. 

If we agree to understand by a first polar to a curve the curve of one 
degree lower which passes through all the points in which the curve is met 
by tangents drawn from an arbitrary point taken anywhere in its own plane, 
we readily perceive that at an ordinary double point all the infinite number 
of first polars which can be drawn to the curve will intersect one another 
at the double point. Again, at a cusp all these polars will not only all 
intersect, they will moreover all touch one another at the cusp. Now we 
may proceed to inquire as to the meaning of a multiplicity of the third degree, 
which, strange to say, I believe has never yet been distinctly assigned by 
geometricians. 

This is not the case of a so-called triple point, that is a point where three 
branches of the curve intersect. Supposing x=0, y = Q,t.o represent such a 
point, the characteristic of the curve must be reducible to the form 

{ga^ + hx'^y + kxy^ -(- ly^) ^""^ + &c., 
which, as is well known, involves the existence of four conditions. This, 
however, would not in itself be at all conclusive against the multiplicity at a 
triple point being only of the third degree ; for it can readily be shown that 
there may exist singular points of any degree of singularity (as measured 
by the number of conditions necessary to be satisfied in order that such 

24—2 



372 On a new Theory of Multiplicity. [46 

singularity may come into existence), but for which the multiplicity may be 
as low as we please ; as, for instance, if at a double point (which is not a cusp) 
there be a point of inflexion on one branch or on both, or a point of undulation, 
or any other singularity whatever, still provided there be no cusps, the 
multiplicity will stick at the first degree and never exceed it ; for only the 
discriminant itself will vanish on these suppositions, but no evectant of the 
discriminant. The reason, on the contrary, why a so-called triple point 
must be said to have a multiplicity of the degree 4, and not merely of the 
degree 3, springs from the fact that the 1st, 2nd and 3rd evectants of the 
discriminant all vanish at such a point. 

It is clear, then, that there ought to exist a species of multiplicity for 
which the 1st and 2nd evectants vanish, but not the 3rd. In fact, as at a 
double point the first polars all merely intersect, but at a cusp have all 
a contact with one another of the first degree, so we ought to expect that 
there should exist a species of multiple point such that all the first polars 
should have with each other a contact of the second degree (or if we like so 
to say, the same curvature) at that point. When the curve has a triple point, 
all its first polars will have that point upon them as a double point ; and it 
is not at the first glance, easy a priori to say what is the nature of the 
contact between two curves which intersect at a point which is a double 
point to each of them : we know upon settled analytical principles, that when 
one curve having a double point is crossed there by another curve not having 
a double point, that the two must be said to have with one another, a contact 
of the 1st degree ; and we now learn from our theory of evection, that if each 
have a double point at the meeting-point, the degree of the contact must 
from principles of analogy be considered to be of the 3rd degree*. Now, then, 
we come to the question of deciding definitely what is a multiple point for 
which the degree of multiplicity is 3. It is, adopting either test, whether 
of first polar contact or of evection, a cusp situated or having its nidus, so to 
say, at a point of inflexion. In other words, a; = 0, y = will be a point 
whose multiplicity is intermediate between that of the cusp and that of 
a so-called triple point, when the characteristic of the curve admits of being 
written under the form 

^11-2^2 -1- z^-^ (gaf + hx^y -\- ixy^) + z"'~* &c. ; 

or in other words, when over and above the vanishing of the constant and 
linear coefficients, and the quadratic coefficient being a perfect square, 
as in the case of an ordinary cusp, this square has a factor in common with 
the next (the cubic) coefficient ; or again, in other words, a curve has a point 

* This may easily be verified by direct analytical means ; as also the more general pro- 
position, that two curves meeting at a point where there are m branches of the one and n 
branches of the other, must be considered to have mn coincident points in common, that is, if we 
like so to express it, to have a contact of the degree mn - 1. 



46] On a new Theory of Multiplicity. 373 

for which the multiplicity is 3 when its characteristic function admits of being 
expanded according to the powers of one of the variables, in such a manner 
that the first coefficient and the second (the linear) coefficient vanish, and 
that the discriminant of the third and the resultant of the third and fourth 
are both at the same time zero. This being the case, it may be shown that 
the first polars will all have with each other a contact of the second degree ; 
and moreover, that all the evectants of the discriminant will have as a 
common factor a linear function of the variables, raised to a power whose 
index is three times that of the characteristic function. As, then, there is but 
one kind of ordinary double point, and but one kind of point with multiplicity 
2, so there is one, and only one, kind of point with a multiplicity 8. A cusp 
is a peculiar double point ; a flex-cusp (as for the moment I call the point 
last above discussed) is a peculiar cusp. This law of unambiguity, howevei-, 
appears to stop at the third degree. A so-called triple point (which ought 
in fact to be called a quintuple point) is a point for which the multiplicity, 
as shown above, is of the fourth degree ; but it is not the only point of that 
degree of multiplicity. Without assuming to have exhausted every possible 
supposition upon which such a degi-ee of multiplicity may be brought into 
existence, it will be sufficient to take as an example a curve whose character- 
istic is capable of assuming the form 

^n-iga ^ ^n-s (^g^ ^ hx'y) + Z'"~* (kof^ + l^^V + Wl«y + UX'lf) + ^''-^ &C. 

It may readily be demonstrated that the first polars of this curve have 
all with one another at the point cc, y a, contact of a degree exceeding the 
2nd, that is of at least the 3rd degree (and, I believe, in general not higher). 
Now the point x, y is evidently not a triple-branched point, but a cusp with 
three additional degrees of singularity ; so that we have evidence of the 
existence of a point whose degree of singularity is 5, and whose multiplicity 
is at least 4, but which is in no sense a modified triple point. It is probably 
true (but to demonstrate this requires a further advance to be made than has 
yet been realized in the theory of the constitution of discriminants) that a 
cusp may be so modified by the nidus at which it is posited, as, without ever 
passing into a triple point, to be capable of furnishing any amount of mul- 
tiplicity whatever, curiously in this contrasting with an ordinary double point, 
no amount whatever of extraordinary singularity imparted to which, or so to 
speak, to its nidus, can ever heighten its multiplicity so as to make it surpass 
the first degree without first converting it into a cusp. I may illustrate the 
nature of a flex-cusp by what happens to a curve of the third degree. When 
it breaks up into a conic and a right line, there are two ordinary double points; 
for the existence of these double points, as for the existence of a cusp, two 
conditions are required. When, however, the right line and conic touch one 
another (a casus omissus this in the works of the special geometers), the 
characters of the cusp and the point of inflexion are combined at the point 



374 On a new Theory of Multiplicity. [46 

of contact ; the multiplicity is of the third degree, and the singularity also 
of a degree not exceeding this ; three conditions only being necessary to 
be satisfied in order that a given cubic may degenerate into such a form ; 
and it will be found that the discriminant and the first and second evectants 
thereof vanish for this case, and that the third evectant of the discriminant 
will be a perfect 9th power ; whereas in order that the cubic may have a 
so-called triple point, that is may degenerate into a trident of diverging rays, 
four conditions must be satisfied, and it will be found that when this is the 
case, the first, second, and third evectants of the discriminant will all vanish, 
and the fourth will be a perfect 12th power of a linear function of the 
variables. I may mention, by the way, at this place, that the law of a 
discriminant and the successive evectants up to the mth inclusive, all 
vanishing, may be expressed otherwise (not in identical, but in equivalent 
or equipollent terms), by saying that the discriminant and all its derivatives 
of a degree not exceeding the with will all vanish — -understanding by a 
derivative of the discriminant any function obtained from the discriminant 
by differentiating it any specified number of times with respect to the 
constants of the function to which it belongs, the same constants being 
repeated or not indifferently*. And very surprising it must be allowed 
to be, stated as a bare analytical fact, that {m + 1) conditions imposed upon 
the coefficients of a function of any number of variables and of any degree 
should suffice to make the inordinately greater number of functions which 
swarm among the derivatives of the mth and inferior degrees of the dis- 
criminant each and all simultaneously vanish. 

Without pushing these observations too far for the patience of the general 
reader, it may be remarked by way of setting foot with our new theory upon 
the almost unvisited region of the singularities of surfaces, that by the light 
of analogy we may proceed with a safe and firm step as far as multiplicity 
of the third degree inclusive. 

The function characteristic of the surface being supposed to be expressed 
in terms of the four variables x, y, z, t, and expanded according to descending 
powers of t, then when x, y, z is an ordinary double point of the first degree 
of multiplicity, the constant and the linear coefficient disappear ; when the 
point has a multiplicity 2, the discriminant of the quadratic coefficient 
will be zero, that is this coefficient will be expressible by means of due linear 
transformations under the form of a;- -f y^ ; and when the multiplicity is to be 
of the degree 3, the cubic coefficient will, at the same time that the quadratic 
coefficient is put under the form x^ -\- y^, itself (for the same system of x and 
y) assume the form of a cubic function of x, y, z, in which the highest power 
of z, that is z^, will not appear ; or in other words (restoring to x, y, z their 

* Or, to speak more simply, the diseriminant and its gucoessive differentials up to the mth 
exclusive must all vanish simultaneously. 



46] On a new Theory of Midtiplicity. 375 

generality), not only will the first derivatives of the quadratic function be 
nullifiable simultaneously with each other, but likewise at the same time 
with the cubic function itself. These three cases will be for surfaces, the 
analogues so far, but only so far as regards the degree of the multiplicity, 
to the double point, cusp, and flex-cusp of curves*. The analogue to the 
so-called triple point of the curves will be a point whose degree of singularity, 
depending upon the vanishing of the six constants in the third coefficient 
(which is a quadratic function of cc, y, z) at the same time as the three 
constants in the linear factor, would seem to be but 6 more than for a double 
point, that is in all 1 -f- 6 or 7, but whose multiplicity, as inferred from 
the nature of the contact of its first polars, which will be of the 7th order, 
would appear to be 8 (a seeming incongruity which I am not at present in a 
condition to explain) f; so that there will apparently be 4 steps of multiplicity 
to interpolate between this case and the case analogous {sub modo) to the 
flex-cusp, last considered. Whether these intervening degrees correspond 
to singularities of an unambiguous kind, no one is at present in a condition 
to offer an opinion. I will conclude with a remark, the result of my experi- 
ence in this kind of inquiry as far as I have yet gone in it, namely that 
it would be most erroneous to regard it as a branch of isolated and merely 
curious or fantastic speculation. Every singularity in a locus corresponds 
to the imposition of certain conditions upon the form of its characteristic ; 
by aid of the theory of evection we are able to connect the existence of these 
conditions with certain consequences happening to the form of the discrimi- 
nant, and thereby it becomes possible, upon known principles of analysis, 
to infer particulars relating to the constitution of the discriminant itself 
in its absolutely general form, very much upon the same principle as when 
the values of a function for particular values of its variable or variables are 
known, the general form of the function thereby itself, to some corresponding 
extent, becomes known. Thus, for instance, I have by the theory of evection 
in its most simple application, been led to a representation of the discriminant 

* At an ordinary conical point of a surface for which the multipHcity is 1, every section 
of the surface is a curve with a double point. When the multiplicity is 2, the cone of contact 
becomes a pair of planes, through the intersection of which any other plane that can be drawn 
cuts the surface in a section having an ordinary cusp of multiplicity 2, but which themselves 
cut the surface in sections, having so-called triple points, so that for these two principal sections 
(which is rather surprising) the multiplicity suddenly jumps up from 2 to 4. All other things 
remaining unaltered when the multiplicity of the conical point is 3, the cusp belonging to any 
section of the surface drawn through any intersection of the two tangent planes passes from an 
ordinary cusp to a flex-cusp. 

t So, too, at a so-called quadruple point in a curve, the degree of the contact of the 1st polars 
is 8, and therefore the multiplicity of the curve at such point is 9 ; but the number of constants 
which vanish for this case (namely all those of the cubic coefficient in x, y) over and above what 
vanish for the case of a so-called triple point is only 4, which is a unit less than the difference 
between the measures of the multiplicities at the respective points ; and this difference continues 
to increase as we pass on to so-caUed quintuple and higher multiple points in the curves. 



376 On a new Theory of Multiplicity. [46 

of a function of two variables under a form very different and very much 
more complete and fecund in consequences than has ever been supposed, 
or than I had myself previously imagined, to be possible. 

According to the opinion expressed by an analyst of the French school, 
of pre-eminent force and sagacity, it is through this theory of multiplicity, 
here for the first time indicated, that we may hope to be able to bridge over 
for the purposes of the highest transcendental analysis, the immense chasm 
which at present separates our knowledge of the intimate constitution of 
functions of two fi'om that of three, or any greater number of variables. 

It is, as I take pleasure in repeating, to a hint from Mr Cayley*, who 
habitually discourses pearls and rubies, that I am indebted for the precious 
and pregnant observation on the form assumed by the first discriminantal 
evectant of a binary function with a pair of equal roots, out of which, 
combined with some antecedent reflections of my own, this new theory of 
multiplicity has taken its rise. The idea of the process of evection, and the 
discovery of its fundamental property of generating what, in my calculus 
of forms {Cambridge and Dublin Mathematical Journal), I have called 
contravariants, is due to my friend M. Hermite. The polar reciprocals of 
curves and other loci are contravariants and, as I have recently succeeded 
in showing, for curves at least, evectants, but of course not discriminantal 
evectants; and I am already able to give the actual explicit rule for the 
formation of the polar reciprocal of curves as high as the 5th degree, which 
with a little labour and consideration can be carried on to the 6th, and in 
fact to curves of any degree n when once we are acquainted with any mode 
of determining all such independent invariants of a function of two variables 
as are of dimensions not exceeding 2 (n — 1) in respect of the coefficients. 

By the special geometers (by whom I mean those who, unvisited by a 
higher inspiration, continue to regard and to cultivate geometry as the 
science of mere sensible space) this problem has only been accomplished, and 
that but recently, for curves whose degrees do not exceed the 4th. Mr Salmon 
has made the happy and brilliant (and by the calculus of forms instantaneously 
demonstrable) discovery, communicated to me in the course of a most 
instructive and suggestive correspondence, that a certain readily ascertainable 

* Mr Cay ley's theorem stood thus :— If 

have two equal roots, and cr be its discriminant, then will 

be a perfect 7ith power. It will easily be seen that this theorem is convertible into a theorem of 
evection by interchanging in the result x and y with y and - x. 



46] On a new Theory of Multiplicity. 2i1^ 

evectant of every discriminant of any function whatever is an exact power of 
its polar reciprocal*. 

I believe that it may be shown, that, with the sole exception of odd- 
degreed functions of two variables, the polar reciprocal itself (as distinguished 
from a power thereof) of every function is an evectant, not (of course) of the 
discriminant, but of some determinable inferior invariant. 

P.S. The terms pluri-simultaneous and pluri-simultaneity, used or 
suggested by me in my last paper in the Magazine, may be advantageously 
replaced by the more euphonious and regularly formed words consimul- 
taneous, consimultaneity. Multiplicity and all its attributes and consequences 
are included as particular cases in the general conception and theory of 
consimultaneity, that is of consimultaneous equations, or, which is the same 
thing, of consimnlevanescent functions. 

* Namely, for a function of degree n, and variability (that is, having a number of variables) 
p, the (n- l)P-'th evect of the discriminant is the (m- l)th power of the polar reciprocal. 



47. 



A DEMONSTRATION OF THE THEOREM THAT EVERY HOMO- 
GENEOUS QUADRATIC POLYNOMIAL IS REDUCIBLE BY 
REAL ORTHOGONAL SUBSTITUTIONS TO THE FORM OF 
A SUM OF POSITIVE AND NEGATIVE SQUARES. 



{Philosophical Magazine, IV. (1852), pp. 138 — 142.] 

It is well known that the reduction of any quadratic polynomial 

(1, l)a;= + 2(l, 2)xy + {2, 2)2/^+... +{n, n)t- 

to the form Ui^^ + ai-rj" + . . . + and", where ^,7)... 6 are linear functions of 
X, y ...t, such that x^ + y^+ ... + t'' remains identical with }^' + ri'^ + ... + 6^ 
(which identity is the characteristic test of orthogonal transformation), 
depends upon the solution of the equation 



(1, 1) + X, (1,2) (l,n) 

(2,1), (2, 2) + X...(2, n) 



(n, 1), 



{n, 2) (n, n) + \ 



= 0. 



The roots of this equation give «!, aj ... a„; and if they are real, it is easily 
shown that the connexions between x,y...t\ ^, rj ... 0, are also real. 
M. Cauchy has somewhere given a proof of the theorem*, that the roots of A. 
in the above equation must necessarily always be real; but the annexed 
demonstration is, I believe, new ; and being very simple, and reposing upon 
a theorem of interest in itself, and capable no doubt of many other applica- 
tions, will, I think, be interesting to the mathematical readers of this 
Magazine. 

* Jacobi and M. Borchardt have also given demonstrations ; that of the latter consists in 
showing that Sturm's functions for ascertaining the total number of real roots expressed by my 
formula (many years ago given in this Magazine) are all, in the case of /(X), representable as the 
sums of squares, and are therefore essentially positive. 



47] 



On Homogeneous Quadratic Polynomials. 



379 



Let 



/W = 



(1, 1) + X, (1,2) (1,m) 

(2, 1), (2, 2) + X (2, 7i) 

(3,1), (3,2), (3, 3)+X...(3, w) 



(«, 1), (n, 2) {n,n)+\ 

it is easily proved that/(X,) x/(— X) 

[1, 1]-\S [1,2] [1,71] 

[2,1], [2, 2]-X=^ ...[2, n] 



[n, 1], [n, 2] [n, 7i]-V 

where [t, e] = (t, 1) x (1, e) + (t, 2) x (2, e)+...+(t, h) x (n, e). 

If, now, for all values of r and s, (r, s) = (s, r), that is, if /(O) becomes the 
complete determinant to a symmetrical matrix, then every term [r, s] in 
the derived matrix becomes a sum of squares, and is essentially positive, 
and (- l)"-f{\) x/(— X) assumes the form 

(X2)» - F (X=)»-i + G {XT""- + ...±L, 

where F, G, ... L will evidently be all positive ; for it may be shown that F 
will be the sum of the squares of the separate terms, that is, of the last 
minor determinants of the given matrix, G the sum of the squares of the 
last but one minors, and so on, L being the square of the complete deter- 
minant. For instance, if 

f(X)= a+X, 7, /3 

7, b + X, a 

13, a, c + X 

-f{X) xf{-X)=X'- F\^ + GX' - H, 

where F=a- + ¥ + c^+2a" + 2^^ + 2f, 

G = {ab- yj + (be - a?y- + (ac - /3"-y 

+ 2 (aa - /3yy +2(bj3- ya)' + 2{cy- a^)^ 

H= a, y, I 

y, b, 

/S, a. 

Hence it follows immediately that/(X) = cannot have imaginary roots; 
for, if possible, let X=p + q v'(— 1), and write 

a-\-p=a', b+p = b', c+p = c', X + p = X', 



380 On Homogeneous Quadratic Polynomials. [47 

y (X) becomes a! + V, 7, /3 

7, &' + X', a 

yS, a, c' + X' 

or say ^ (X'), and the equation (^ (X') x ^ (— X') = will be of the form 

X'«-jP'X'^ + (?'X'2-5"' = 0, 
where F' , G', H' are all essentially positive. Hence, by Descartes' rule, no 
value of X'= can be negative, that is, (\—p)- cannot be of the form —q-; 
that is to say, it is impossible for any of the roots of /(X) = to be imaginary, 
or, as was to be demonstrated, all the roots are real. 

I may take this occasion to remark, that by whatever linear substitutions, 
orthogonal or otherwise, a given polynomial be reduced to the form 2^if", 
the number of positive and negative coefficients is invariable : this is easily 
proved. If now we proceed to reduce the form (expressed under the umbral 
notation) (aiOCi + a^x^ + ... + anOCnf to the form 

by first driving out the mixed terms in which x-y enters, then those in which 
x^ enters, and so forth until eventually only a;„ of the original variables is 
left, it may readily be shown that 



'■=(:)■ 






Xa^a^ ...aj 



faiao....an-i\ 
\a1a2 ... an-J ' 

It follows, therefore, that in whatever order we arrange the umbrae ajOa 
the number of variations and of continuations of sign in the series 



^ f<h.\ fellah's /aia.,...an\ 
' \aj ' Ka^a^) ' ' ' Ka^az . . . aJ ' 



will be invariable, and in fact will be the same as the number of positive 
and negative roots in the generating function in X above treated of, that is, 
since all the roots are real, will be the same as the number of variations 
and continuations in the series formed by the coefficients of the several 
powers of X, that is 






1 5"f"'M ■s;'/"l"'2\ jClldl' 



The first part of this theorem admits of an easy direct demonstration ; 
for by my theory of compound determinants, given in this Magazine*, we 
know that 

Oi ffla . . . ar-i ar\ a^a^... a^-^ a^+J 

tti tta . . . ar-1 a^ a-ya„ ... ar-i ctr+i 

Oia^ . . . ttr-A (a^az . . . ar-i«rCJ)-+A 

a^a-i ... ar-J Ka-^a^ ... ar-iarttr+J ' 

[* Cf. pp. 241, 252 above.] 



47] On Homogeneous Quadratic Polynomials. 381 

The first member of this equation is equivalent to 

/tti as . . . ttr-i aA (o-i ^2 • • • ctr-i ar+l^ _fa^az... a^_i a^ V 

Veil eta . . . ttr-i flj-/ \a, ffij . . . a^_i a,+i / Vcti Clz • • • Cir-i Ctr+i '' 

Hence it follows, that if the two factors on the right-hand side of the 
equation have the same sign, 



tti a^... Ur-i dr' Vdi 0^2 • • • Ctr-l C*r+1, 



have also the same sign inter se, and consequently the two triads 
Va^ai . . . ar-il faiaa . . . a^-idr 1 foiaa . . . ar^iarar+il 

[aia^ . . . CSr-lJ ' L^^'l^a • • • «'r^l'*>-J ' L'^l'^2 • • • Cfr-iarCfy+lJ ' 

, fai as . . . Or-il Fai tta . . . a^_i a,+il Fai ^2 • • • «»■- 1 f^r+i ar"! 

[_ai 0.2 . . . Or-i J ' L*^ 0.2 .. . ay_i Or+i J ' L'^i eta • • • Ctr-i O^r+i Ctr J ' 

will in all cases present the same number of changes and continuations, 
which proves that the contiguous umbrae, a^, o^+i, may be interchanged 
without affecting the number of variations and continuations in the entire 
series ; but, as is well known, any one order of elements is always convertible 
into any other order by means of successive interchanges of contiguous 
elements, which demonstrates that, in whatever order the elements Oj, Oj-.-a^ 
be arranged, the number of continuations and variations in 

OjOjA /OjOa ••• (in\ 



I, r 

is invariable. But that the same thing is true (as we know it to be), for the 
relation between any one of these unsymmetrical series and the symmetrical 
series (resulting from the method of orthogonal transformation) 

' \aj' ~' \aiaj ' '" Ka^a^. 
is by no means so easily demonstrable in the general case by a direct method, 
and the attention of algebraists is invited to supply such direct method of 
demonstration. My knowledge of the fact of this equivalence is, as I have 
stated, deduced from that remarkable but simple law to which I have 
adverted, which affirms the invariability of the number of the positive and 
negative signs between all linearly equivalent functions of the form 2 + c^«'' 
(subject, of course, to the condition that the equivalence is expressible by 
means of equations into which only real quantities enter) ; a law to which 
my view of the physical meaning of quantity of matter inclines me, upon the 
ground of analogy, to give the name of the Law of Inertia for Quadratic 
Forms, as expressing the fact of the existence of an invariable number 
inseparably attached to such forms. 



48. 



ON STAUDT'S THEOREMS CONCERNING THE CONTENTS OF 
POLYGONS AND POLYHEDRONS, WITH A NOTE ON A 
NEW AND RESEMBLING CLASS OF THEOREMS. 

[Philosophical Magazine, IV. (1852), pp. 335 — 345.] 

The beautiful and important geometrical theorems of Staudt are, I 
believe, little, if at all, known to English mathematicians. They originally- 
appeared in Crelles Journal for the year 1843, and have been recently 
reproduced in M. Terquem's Kouvelles Annales for the August Number of 
the present year. 

These theorems may be summed up, in a word, as intended to show the 
possibility and method of expressing the product of any two polygons or any 
two polyhedrons as entire functions of the squares of the distances of the 
angular points of the two figures from one another. The well-known expres- 
sion for the square of the area of a triangle in terms of the sides (in which, 
when expanded, only even powers of the lengths of the sides appear), is but 
a particular case of Staudt's theorem for polygons, for it may be considered 
as the case of two equal and similar triangles whose angular points coincide. 
So in like manner, as observed by Staudt, a similar expression in terms of 
its sides may be found for the square of a pyramid. This expression had, 
however, been previously given (although, by a strangle negligence, not 
named for what it was) by Mr Cayley in the Cambridge Mathematical 
Journal for the year 1841*, in his paper on the relations between the 
mutual distances to one another of four points in a plane and five points in 
space ; the singularly ingenious (and as singularly undisclosed) principle of 
that paper consisting in obtaining an expression for the volume of a pyramid 
in terms of its sides, and equating this, or rather its square, to zero as the 
conditions of the four angular points lying in the same plane. 

* Query, Is not this expression for the volume of a pyramid in terms of its sides to be found 
in some previous writer ? It can hardly have escaped inquiry. 



\ 



48] 



On Staudt's Theorems. 



383 



The analogous condition for five points in space is virtually deduced by 
going out into rational space of four dimensions, and equating to zero the 
expression obtained for the volume of a plupyramid ; meaning thereby the 
figure which stands in the same relation to space of four as a pyramid to 
space of three dimensions. Mr Cayley's method, if it had been pursued 
a step further, would have led him to a complete anticipation of the principal 
part of Staudt's discovery. The method here given is not substantially 
different from Mr Cayley's, but is made to rest upon a more general 
principle of transformation than that which he has employed. As to 
Staudt's own method, it is as clumsy and circuitous as his results are simple 
and beautiful. Geometry, trigonometry and statics, are laid under contri- 
bution to demonstrate relations which will be seen to flow as immediate and 
obvious consequences from the most elementary principles in the algorithm 
of determinants. Perhaps, however, M. Staudt's method is as good as could 
be found in the absence of the application of the method of determinants, 
the powers of which, even so recently as ten years ago, were not so well 
understood or so freely applied as at the present day. 

The following new but simple theorem, of which I shall have occasion to 
make use, will be found to be a very useful addition to the ordinary method 
for the multiplication of determinants. "If the determinants represented 
by two square matrices are to be multiplied together, any number of columns 
may he cut off from the one matrix, and a corresponding number of columns 
from the other. Each of the lines in either one of the matrices so reduced 
in width as aforesaid being then multiplied by each line of the other, and 
the results of the multiplication arranged as a square matrix and bordered 
with the two respective sets of columns cut off arranged symmetrically (the 
one set parallel to the new columns, the other set parallel to the new lines), 
the complete determinant reP'- jented by the new matrix so bordered 
(abstraction made of the algebraical sign) will be the product of the two 
original determinants." 

Thus ( 7 ) X ( ^ J may be put under any one of the three following forms : — 





aOL-lr 


b^, oy + bB 1 






ca + d/3, cy + d8\ 




aa, 


ay, b 




2, 2, a, 


b 


COL, 


cy, d 


or 


2, 2, c, 


d 


/3, 


S, 




a, /3, 0, 
7, S, 0, 







* Any quantities might be substituted instead of 2 in tbe places occupied by the figure in the 
above determinant, as such terms do not influence the result ; this figure is probably, however, 
the proper quantity arising from the application of the rule, because (as all who have calculated 
with determinants are aware) the value of the determinant represented by a matrix of no places 
is not zero but unity. 



384 



On Staudt's Theorems concerning the 



[48 



And in general for two matrices of ri^ terms each, this rule of multiplication 
will give (w + 1) distinct forms representing their products. 

Thus, as a further example, 

a, b, I 



a, 



h', 
b", 



a, /8, 



besides the first and last forms, will be representable by the two intermediate 
forms 

aa + &/3, aa + 6/3', aa" + bS", 

a'a + 6'/3, a'a + b'^', a'a" + b'/3", 

a"oL + 6"/S, a'V + 6"/3', a'a!' + 6"/3", 



and 







aa, aa. , 

a'a, act', 

a"a, a"a, 

/3, /3', 

7. 7. 



aa 
a'a" 
a" a!' 



b, 
b', 
b", 
0, 
0, 



To arrive, for instance, at the latter of these two forms, we have only 
to write the two given matrices under the respective forms 



b, c, 0, 

b', c', 0, 

b", c", 0, 

0, 0, 1, 

0, 0, 0, 1 



a, 0, 0, ^, 

a', 0, 0, /3', 

a", 0, 0, yS", 

0, 1, 0, 0, 

0, 0, 1, 0, 



and then apply the ordinary rule of multiplication. So, again, to arrive 
at the first of the above written two forms, we must write the two given 
matrices under the respective forms 

a, b, c, a, ^, 0, 7 

a', b', c', a', /3', 0, 7' 

„ ,,/ „ « and — „ ^„ „ 
a", b' , c' , a , ^', 0, 7" 

0, 0, 0, 1 0, 0, 1, 

and proceed as before. 

This rule is interesting as exhibiting, as above shown, a complete scale 
whereby we may descend from the ordinary mode of representing the product 
of two determinants to the form, also known, where the two original deter- 



48] 



contents of Polygons and Polyhedrons. 



385 



minants are made to occupy opposite quadrants of a square whose places 
in one of the remaining quadrants are left vacant, and shows us that under 
one aspect at least this latter form may be regarded as a matrix bordered 
by the two given matrices. 

A second but obvious theorem requiring preliminary notice is the 
following, namely that the value of the determinant to the matrix 

'^l,!; '^1,2 ••• ^,nt -I) 

^2,1 J ^2,2 ••• ^2, 71 J J-J 



C^n, 1 ) 0,11,1 ••• ^n, n > 


1, 


1, 1, ... 1, 


0, 


is the same as the value of the determinant to the matrix 


^,,1, A,^„... A,^n, 


1, 


■^2,1) -42,2 ••• -^S.Jl) 


1, 


-4n,i, An^^i ... An,n 


1, 


1, 1, ... 1, 


0, 



where in general 

hijh^ ... h„ and ki,kc^...kn being any two perfectly arbitrary series of quantities. 
This simple transformation is of course derived by adding to the respective 
columns in the first matrix the last column (consisting of units) multiplied 
respectively by h^, h2 ... hn, 0; and to the respective lines, the last line 
(consisting of units) multiplied respectively by k^, k^ ... k,i, 0. 

Suppose, now, that we have two tetrahedrons whose volumes are repre- 
sented respectively by one-sixth of the respective determinants 



yi> 


^1, 


1 


, 


?1, 


Vi, 


ri. 


1 


2/2. 


^2, 


1 




^2, 


V2, 


?2, 


1 


2/3, 


Zl, 


1 




?3, 


Vi, 


?3, 


1 


2/4- 


Zi, 


1 




^4, 


V4, 


?4, 


1 



^r, 1/r, ^r representing the orthogonal coordinates of the point r in one 
tetrahedron, and ^„ rj^, ^r the same for the point r in the other. 

By the first theorem their product may be represented (striking off the 
last column only from each matrix) by the matrix 

SiTifi, 2a!i^2, 2a;i^3, Sa^^j, 1 

S^al^l, 2«2^2. 2a;2^3. S«2^4, 1 

-^^351, •^'^352? — ^3S3j -^^354, ^ 

XsCi^i, Sxi^i, 2a;4^3, 2a;4^4, 1 

1, 1, 1, 1, 

25 



386 On StaudCs Theorems concerning the [48 

where, in general, any such term as S^^fs represents 

Again, by virtue of the second theorem, adding 

-l^x,\ -i2«/, -i2«3^ -iS^/ 
to the respective lines, and 

-^2^ -iS?.^ -hn^; -^Sf/' 

to the respective columns, the above matrix becomes (after a change of signs 
not affecting the result) the — |th of 

^{^i-h)\ 2(a:,-r=n 2(^1- ^3)=. 2 (a;, -?,)=, 1 

2(a;.-?i)^ ^{x,-^,r, ^{x,-^,f, ^{x,-^,Y. 1 

2(*3-^i)^ 2(^3-f.)^ ^{cc,-h)\ 2(^3-^4^ 1 

2(*'.-^0% 2(^,-f,)^ 2(^,-^3)^ 2(0^,-^4)^ 1 

1, 1, 1, 1, 

or calling the angular points of the one tetrahedron a, b, c, d, and of the 
other p, q, r, s, 8 x 36, that is 288 times, their product is representable by 
— 1 X the determinant 

{ap)-, {aqf, (arf, (as)", 1 

(bpY, {hqf, {hrf, {hsf, 1 

{cpY, {cqf, {cry, {csf, 1 

{dpy, {dqf, (dry, (dsy, 1 

1, 1, 1, 1, 

and of course if p, q, r, s coincide respectively with a, b, c, d, 576 times the 
square of the tetrahedron abed will be represented under Mr Cayley's form, 

0, (aby, (acy, (ady, 1 
(bay, 0, (bey, (bdy, 1 
{cay, {cby, 0, (cdy, 1 
{day, {dby, {dcy, 0, 1 

1, 1, 1, 1, , 

four out of the sixteen distances vanishing, and the remaining twelve 
reducing to six pairs of equal distances. The demonstration of Staudt's 



* The corresponding quantity to the above determinant for the case of the triangle (hereafter 
given) is identical with the Norm to the sum of the sides. I have succeeded in finding the 
Factor (of ten dimensions in respect of the edges), which, multiplied by the above Determinant 
itself, expresses the Norm to the sum of the Faces, that is, the superficial area of the Tetrahedron. 



48] 



contents of Polygons and Polyhedrons. 



387 



theorem for triangles is obtained in precisely the same way by throwing the 
product of the two determinants 







«i, Vi, 1 




?., Vu 1 






x^, 2/2, 1 


and 


?2, -^2, 1 






X3, 2/3. 1 




Is, %, 1 




under the form of — ;Jth of 






1{w,-^,)\ t{x,-^,f, 


2(^l-f3)', 1 




2(a;.-?0^ 2(03,-^2?, 


^{x.-^,r, 1 




^{x^-^.y, 2(*s-i^.n 


2(0.3-13^ 1 






1, 


1, 


1, 






When the two triangles coincide, calling their angular points a, b, c 
the above written determinant becomes 



0, 


(aby, 


(acy, 


1 


{bay, 


0, 


(bey, 


1 


(cay, 


(cby, 


0, 


1 


1, 


1, 


1, 





(aby + (acY + (bey - 2 {aby . (acy - 2 (aby . (bey - 2 (acy . (bey, 

the negative of which is the well-known form expressing the square of four 
times the area of the triangle abc. 

There is another and more general theorem of Staudt for two triangles 
not in the same plane, which may be obtained with equal facility. In fact, 
if we start from the determinant 

(any, (a^y, (a^y, 1 
(bay, (b/sy, (byy, 1 
(cay, (cl3y, (cyY, 1 

1, 1, 1, 

and add to each column respectively the last column multiplied by e^i', e^^, 
e^s respectively, we arrive at the form 

(aay+e^^\ (a^y + e^i, (a^y+e^, 1 

(bay + e^,\ (b/3y+e^,\ (b^y + e^, 1 

(cay+e^,\ (c^y + e^^, (cjY + e^,% 1 

1, 1, 1, 

and considering |i, 771; ^2, t],^', Is, % as the coordinates of a, /8, 7, the 

25—2 



388 On Staudt's Theorems concerning the [48 

projections upon the plane of abc of a triangle ABC, whose plane intersects 
the former plane in the axis of y, and makes with that plane an angle whose 
tangent is e, it is easily seen that this determinant is term for term identical 
with the determinant 



(aAy, 


(aBy, 


{aG)\ 


1 


(bAy, 


{bBf, 


{hC)\ 


1 


(cAY, 


{cBf, 


(ccy, 


1 


1, 


1, 


1, 






which therefore expresses — 16 times the product of the triangles abc and 
a^7, that is abc x ABC x cosine of the angle between the two. A similar 
method, if we ascend from sensible to rational geometry, may be given for 
expressing in terms of the distances the product of any two pyramids (in 
a hyperspace) by the cosine of the angle included between the two infinite 
spaces * in which they respectively lie. To pass from the cases which have 
been considered of two triangles to two polygons, or of two tetrahedrons to 
two polyhedrons, generally presents no difficulty ; and for Professor Staudt's 
method of doing so, which is simple and ingenious, and does not admit of 
material improvement, the reader is referred to the memoir in Crelle's Journal 
or Terquem's Annales already adverted to. It is, however, to be remarked 
(and this does not appear to be sufficiently noticed in the memoirs referred 
to), that whilst the expression for the product of any two polygons in terms 
of the distances given by Staudt's theorem is unique, that for the product 
of two polyhedrons given by the same is not so, but will admit of as many 
varieties of representation as there are units in the product of the numbers 
respectively expressing the number of ways in which each polygonal face of 
each polyhedron admits of being mapped out into triangles. I cannot help 
conjecturing (and it is to be wished that Professor Staudt or some other 
geometrician would consider this point) that in every case there exists, 
linearly derivable from Staudt's optional formulae (but not coincident with 
any one of them), some unique and best, because most symmetrical, formula 
for expressing the product of two polyhedrons in terms of the distances of 
the angular points of the one from those of the other. In conclusion I may 
observe, that there is a theorem for distances measured on a given straight 
line, which, although not mentioned by Staudt, belongs to precisely the same 
class as his theorems for areas in a plane and volumes in space; namely 
a theorem which expresses twice the rectangle of any two such distances 
under the form of an aggregate of four squares, two taken positively and two 

* In rational or uniYersal geometry, that which is commonly termed infinite space (as if it 
were something absolute and unique, and to which, by the conditions of our being, the repre- 
sentative power of the understanding is limited), is regarded as a single homaloid related to a 
plane, precisely in the same way as a plane is to a right line. Universal geometry brings home 
to the mind with an irresistible force of conviction the truth of the Kantian doctrine of locality. 



48] 



contents of Polygons and Polyhedrons. 



389 



negatively; that is to say, if A, B, G, D be any four points on a right line 
2ABxCD = AI)' + B(?-AC'-BD\ I know not whether this theorem 
be new, but it is one which evidently must be of considerable utility to the 
practical geometer. 



Note on the above. 

The fundamental theorem in determinants, published by me in the 
Philosophical Magazine in the course of last year*, leads immediately to a 
class of theorems strongly resembling, and doubtless intimately connected 
with, those of Staudt. 

Thus for triangles we have by this fundamental theorem 

V3 
I 



2/2, 2/3 

1, 1 

Vi> Vi 

1, 1 

|3, ?1 

Vs, Vi 

1, 1 



1. 1. 

£3, ^21 ^3 

%, 2/2, 2/3 

1, 1, 1 

Vi, 2/2> 2/3 

1, 1, 1 



«1, 


?2, 


I3 




1:. 


^2, 


OSi 


yi> 


V2, 


% 


X 


%> 


2/2. 


2/3 


1, 


1, 


1 




1, 


1, 


1 



and consequently, if ABC, DEF be any two triangles, 

ABC X DEF= ABE x FBG + AEF x DBC+AFD x ^C^^. 

This may be considered a theorem relating to two ternary systems of 
points in a plane. The analogous and similarly obtainable theorem for two 
binary systems of points in the same right line is 

ABxGD = ACxDB-ADxCB. 

As in applying this last theorem to obtain correct numerical results we must 
give the same algebraical sign to any two lengths denoted by the two 
arrangements XY, ZT, according as the direction from X to F is the same 
as that from Z to T, or contrary to it, so in the theorem for the products 
of triangles, the areas denoted by any two ternary arrangements XYZ, TUV 
must be taken with the like or the contrary sign, according as the direction 
of the rotation XYZ is consentient ■n'ith or contrary to that of TUV; so 
that three of the six possible arrangements of XYZ may be used indifferently 
for one another, but the other three would imply a change of sign. If we 
[* See pp. 249, 253 above.] 



390 



On Staudt's Theorems concerning the 



[48 



analyse what we mean by fixing the direction of the rotation of XYZ, and 
reduce this form of speech to its simplest terms, we easily see that it amounts 
to ascertaining on which side of B, C lies, that is whether to its right or left, 
to a spectator stationed at J. on a given side of the plane ABC. 

Let us now pass to the corresponding theorems for two tetrahedrons put 
respectively under the forms 

r:- r.. Is. ?4 



2/1. 2/2, 2/3, 2/4 

1, 1 



Vu V2, Va, Vi 

?u r=, rs, r. 



1, 1, 1, 1 1, 1, 1, 

We may represent this product in either of two ways by the application 
of our fundamental theorem, namely as 



2/1. 


Vi, 


V2, 


•2:. 


?1, 


r., 


1, 


1, 


1, 


X,, 


«2, 


ii. 


2/1. 


2/ii> 


J?!. 


Zl, 


Zl, 


ri, 


1, 


1, 


1, 



^4, 


^2, 


'74, 


2/=, 


r4, 


^2, 


1, 


1, 


Is, 


I4, 


%, 


Vi, 


fs, 


?4, 


1, 


1, 



2/3, 2/4 



ys, 2/4 



+ &c. 



+ &C. 



there being four products to be added together in the first expression and 
six in the latter; and the rule, if we wish that all the products may be 
additive, being that on removing the sign of multiplication the determinant 
to the square matrix formed by the Greek letters in situ shall always preserve 
the same sign. Hence we derive two geometrical formulie concerning the 
products of polyhedrons, namely 

(1) ABCD X EFGH = ABCJE x FGRD - ABGF x GHED 

+ ABGG X EEFD - ABGH x FGED. 

(2) ABGD x EFGH = ABEF x GHCD + ABGH x EFGD 

+ ABEG X HFGD + ABHF x EGCD 
+ABEHX FGGD + ABFG x EHGD. 

These formulae give rise to an exceedingly interesting observation. In 
order that they shall be numerically true, we must have a rule for fixing 
the sign to be given to the solid content represented by any reading off of 
the four points of a tetrahedron, that is we must have a rule for determining 



48] contents of Polygons and Polyhedrons. 391 

the sign of solid contents of figures situated anywhere in space analogous 
to that which, as applied to linear distances reckoned on a given right line, 
is the true foundation of the language of trigonometry, and the condition 
precedent for the possibility of any system of analytical geometry such as 
exists, and which, not altogether without surprise, I have observed in the 
pages of this Magazine one of the learned contributors has thought it necessary 
to vindicate the propriety of importing into his theory of quaternions. 

Various rules may be given for fixing the sign of a tetrahedron denoted 
by a given order of four letters. One is the following : the content of A BCD 
is to be taken positive or negative, according as to a spectator at A the 
rotation of BCD is positive or negative. Another, again, is to consider AB 
and CD as representing, say two electrical currents, and to suppose a spectator 
so placed that the current AB shall pass through the longitudinal axis of his 
body from the head towards the feet, and looking towards the other current 
CD ; the sign of the solid content of the tetrahedron (and, indeed, also the 
effect, in a general sense, of the action of the two currents upon one another) 
will depend upon the circumstance of this latter current appearing to flow 
from the right to the left, or contrariwise in respect of the spectator. Last 
and simplest mode of all, the sign of the solid content of ABCD will depend 
upon the nature (in respect to its being a right-handed or left-handed-screw) 
of any regular screw-line (whether the common helix or one in which the 
increase or decrease of the inclination is always in the same direction) 
terminating at B and C, and so taken that BA shall be the direction of the 
tangent produced at B, and CD the direction of the tangent produced at C. 
Inasmuch as of the twenty-four permutations of a quaternary arrangement 
a defined twelve have one sign, and the other twelve the contrary sign, these 
various definitions of the direction, or, as it may be termed, polarity, of a 
tetrahedron corresponding to a given reading, whether as taken each in itself 
or compared one with another, give rise to, or rather imply a considerable 
number of interesting theorems included in our intuitions of space, and 
probably belonging to the, in my belief, inexhaustible class of primary and 
indemonstrable truths of the understanding. 



49. 



ON A SIMPLE GEOMETRICAL PROBLEM ILLUSTRATING A 
CONJECTURED PRINCIPLE IN THE THEORY OF GEO- 
METRICAL METHOD. 

[Philosophical Magazine, IV. (1852), pp. 366 — 369.] 

The following theorem deserves attention as illustrating a principle of 
geometrical method which will be presently adverted to. It is curious, also, 
from the fact of its solution being by no means so obvious and self-evident 
as one would expect from the extreme simplicity of its enunciation. It 
appeared, and for the first time, it is believed, at the University of Cambridge 
about a twelvemonth back, where it excited considerable attention among 
some of the mathematicians of the place. The proposition, as originally 
presented, was merely to prove that if ABG be a triangle, and if AD and 
BE drawn bisecting the angles at A and B and meeting the opposite sides 
in D and E be equal, then the triangle must be isosceles. It is particularly 




noticeable that all the geometrical demonstrations yet given of this theorem 
are indirect. Thus the first and simplest (communicated to me by a promising 
young geometrician, Mr B. L. Smith of Jesus College, Cambridge), was the 
following : — Assume one of the angles at DAB to be greater than the corre- 
sponding angle EBA ; it can easily be shown that, upon this supposition, 
D will be higher up from AB than E ; so that if DF and EG be drawn 
parallel to AB, DF will be above EG ; it is then easily shown that DF= AF, 
EG=BG, and consequently DF and AF are each respectively less than EG 



49] On a simple Geometrical Problem. 393 

and BG; and also DFA, which is the supplement of twice DAB, will be less 
than EGB, which is the supplement of twice FBA ; from which it is readily 
inferred, by an easy corollary to a proposition of Euclid, that DA will be less 
than FB, whereas it should be equal to it ; so that neither of the half angles 
at the base can be greater than the other, and the triangle is proved to be 
isosceles. Another and independent demonstration by the writer of this 
article is less simple, but has the advantage of lending itself at once to a 
considerable generalization of the theorem as proposed. Assuming, as above, 
that DAB is greater than EBA, it is easily seen that Zl^" produced will cut 
BA at K on the side of it : also ii AD and BE intersect in H, it is readily 
demonstrable, by a suitably constructed apparatus of similar triangles, that 

AH:BH::CE:CD. 

But as HBA is less than HAB, AH is less than BH, and therefore 
CE is less than CD, and therefore CED is greater than CDE; that is to say, 
CAB less K is greater than GBA plus K, and therefore DAB less K is 
greater than EBA, that is ADE is greater than ABE, and therefore the 
perpendicular from A upon DE is greater than that from E on AB, which 
is easily proved to be absurd. Hence, as before, the triangle is proved to be 
isosceles. This proof, it is obvious, remains good for all cases in which EB 
and DA, drawn on either side of the base, divide the angles at the base 
proportionally, provided that these lines remain equal, and make positive 
or negative angles with the base not less than one-half of the respective 
corresponding angles which the sides of the triangle are supposed to make 
with it. The analytical solution of the question, as might be expected, 
extends the result still further. To obtain this, let 

BAG = n. BAD, ABG = n.ABE, 

n for the present being any numerical quantity, positive or negative ; 
calling BAC=2na, ABG=2n/3, we readily obtain, by comparison of the 
equal dividing lines with the base of the triangle, 

sin {2na + 2/3) _ sin (2??./3 + 2a) 
sin 2na sin 2«/3 ' 

sin (2na + 2/3) _ sin 2na 
sin (2)1/3 + 2a) sin 2??/3 ' 

and by an obvious reduction, 

tan (n - 1) (a - /3) _ tan (n + 1) (a + /3) 
tan n (a — ^) tan n (a + /3) 

When this equation is put under an integer form, it is of course satisfied 
by making a = /3 ; on any other supposition than a = /8 it evidently cannot 
be satisfied by admissible values of the angles for any value of n between 



394 On a simple Geometrical Problem. [49 

+ 1 and + 00 ; for on that supposition, since (a — ^) and (a + /3) are each less 

180 
than — — , the first side of the equation will be necessarily a proper fraction 

and positive; but the second side, either a positive improper fraction if 
(n + !)(« + /3) be less, and a negative proper or a negative improper fraction 
if (n + 1) (a + y8) be greater than a right angle. 

If n be negative, let it equal — v, then 

tan(i; + l)(a-;e) _ tan (y - 1) (a + /3) 
tan v(a — ^) tan v (a + /8) ' 

and for the same reason as before, if v lies between oo and 1, this equation 
cannot be satisfied. Hence the theorem is proved to be true for all values 
of n, except between + 1 and — 1. For these values it ceases to be true ; 
in fact, for such values for any given values of (a — y8) there will be always, 
as it may be easily proved, one or more values of (a + /8) ; thus if n = \, the 
equation becomes 



and if n = — i , 



tan 3 , 
tan • — - — 



Un'sC"-^ 



-=-1: 



= -1, 



showing that a +,8 =90 and a — /8 = +90 in these respective cases will 
afford a solution over and above the solution « = /S, which is easily verified 
geometrically*. It would be an interesting inquiry (for those who have 
leisure for such investigations) to determine for any given value of n between 
+ 1 and — 1 the superior and inferior limits to the number of admissible values 
of a + /3 corresponding to any given value of a — /3+. 

My reader will now be prepared to see why it is that all the geometrical 
demonstrations given of this theorem, even in the simplest case of all, namely 
when n = 2, are indirect, I believe I may venture to say necessarily indirect. 
It is because the truth of the theorem depends on the necessary non-existence 
of real roots (between prescribed limits) of the analytical equation expressing 
the conditions of the question; and I believe that it may be safely taken 
as an axiom in geometrical method, that whenever this is the case no other 

* In the first of these cases, if the base of the triangle is supposed given, the locus of the 
vertex is a right line and a circle ; in the second case, a right line and an equilateral hyperbola. 

t When ±n lies between ^ — - and ^ — - (i being any positive integer), it is easily seen that 

the superior limit must be at least as great as i. 



49] On a simple Geometrical Problem. 395 

form of proof than that of the reductio ad ahsurdum is possible in the nature 
of things. If this principle is erroneous, it must admit of an easy refutation 
in particular instances. 

As an example, I throw out (not a challenge, but) an invitation to discover 
a direct proof, if such exist, of the following geometrical theorem, as simple 
a one as it is perhaps possible to imagine : — " To prove that if from the 
middle of a circular arc two chords be drawn, and the remoter segments 
of these chords cut off by the line joining the end of the arc be equal, the 
nearer segments will also be equal." The analytical proof depends upon the 
fact of the equation x^ -\-ax = 6^ (where a is the given length of each segment, 
and b the length of the chord of half the given arc) having only one admis- 
sible root ; and if the principle assumed or presumed to be true be valid, no 
other form of pure geometrical demonstration than the reductio ad ahsurdum 
should be applicable in this case. For the converse case, where the nearer 
segments are given equal, the reducing equation is a(a+a;) = 6^ indicating 
nothing to the contrary of the possibility of there being a direct solution, 
which accordingly is easilj' shown to exist. The indirect form of demonstra- 
tion, it may be mentioned, is sometimes liable to be introduced in a manner 
to escape notice. As, for instance, if it should be taken for granted in the 
course of an argument, that one triangle upon the same base and the same 
side of it as another triangle, and having the same vertical angle, must have 
its vertex lying on the same arc ; this would seem to be immediately true by 
virtue of the well-known theorem, that angles in the same circular segment 
are equal, but in reality can only be inferred from it indirectly by showing 
the impossibility of its lying outside or inside the arc in question. To go one 
step further, I believe it to be the case, that granted to be true all those 
fundamental propositions in geometry which are presupposed in the principles 
upon which the language of analytical geometry is constructed, then that the 
reductio ad ahsurdum not only is of necessity to be employed, but moreover 
in propositions of an affirmative chai-acter never need be employed, except 
when as above explained the analytical demonstration is founded on the 
impossibility or inadmissibility of certain roots due to the degree of the 
equation implied in the conditions of the question. If this surmise turn out 
to be correct, we are furnished with a universal criterion for determining 
when the use of the indirect method of geometrical proof should he considered 
valid and admissible and when not*. 

* If report may be believed, intellects capable of extending the bounds of the planetary 
system and lighting up new regions of the universe with the torch of analysis, have been bafSed 
by the difficulties of the elementary problem stated at the outset of this paper, in consequence, 
it is to be presumed, of seeking a form of geometrical demonstration of which the question from 
its nature does not admit. If this be so, no better evidence could be desired to evince the 
importance of such a criterion as that suggested in the text. 



50. 



ON THE EXPRESSIONS FOR THE QUOTIENTS WHICH APPEAR 
IN THE APPLICATION OF STURM'S METHOD TO THE 
DISCOVERY OF THE REAL ROOTS OF AN EQUATION. 

[Hull British Association Report (1853), Part ii., pp. 1—3.] 

Many years ago I published expressions for the residues which appear in 
the application of the process of common measure to fx and f'x, and which 
constitute Sturm's auxiliary functions. These expressions are complete 
functions of the factors of foo and of differences of the roots of fx, and are 
therefore in effect functions of the factors exclusively, since the difference 
between any two roots may be expressed as the difference between two 
corresponding factors. Having found that in the practical applications of 
Sturm's theorem the quotients may be employed with advantage to replace 
the use of the residues, I have been led to consider their constitution ; and 
having succeeded in expressing these quotients (which are of course linear 
functions of x) under a similar form to that of the residues, that is, as 
complete functions of the factors and differences of the I'oots of fx, I have 
pleasure in submitting the result to the notice of the Mathematical Section 
of the British Association. 

Let h^, hr,, ha ... /i„ be the n roots offx. 

Let ^(a,b,c...l) in general denote the squared product of the differences 
oi a,b, c ...I. 

Let Zi denote in general 'Z^{hg^he^... hg.), where 6i, 62 -..Oi indicate any 
combination of i out of the n quantities a,b,c,... I, with the convention that 
Zo = l, Z^ = n; and let (i) denote ^ {1 + (— 1)'}, being zero when i is odd, and 
unity when i is even ; then I find that the ^'th quotient Qi may be written 
under the form 

Qi = iPl= (« - k) + iP,' (x-h)+...+ iPn' {cc - A„), 

where in general 



iP. 



Zi_, Z\_, Z\_, 



Zi Z-i-2 Z\^i Z'\i)+i 
X 2 {UKM ■■■ KJ X (k - K) i'^e - he.)... {he-he_,)]. 



60] Oil Sturm's Method of Real Roots of an Equation. 397 

f'x 
If we suppose -j—, by means of the common measure process, to be 

expanded under the form of an improper continued fraction, the successive 
quotients will be the values of Qi, Qj ... Q„ above found, that is 

/^_ J 1 ]_ J^. 

fx ~ Qi- Q2- Q3- '" Qn 

the successive convergents of this fraction will be 

1 Q. Q2Q3-I f'«= 



The numerators and denominators of these convergents will consequently 
also be functions of the factors exclusively. They are the quantities the sum 
of the products of which multiplied respectively by foo and f'x produce (to 
constant factors pris) the residues. The denominators are expressible very 
simply in terms of the factors and the differences of the roots ; and their 
values under such forms were published by me about the same time as the 
values of the residues in the Philosophical Magazine; the expression for 
the numerators is much more complicated, but is given in my paper, " The 
Syzygetic Relations," &c., in the Philosophical Transactions, [p. 429 below.] 

By comparing the expression for any quotient with the expressions for 
the two residues from which it may be derived, we obtain the following 
remarkable identity : Zi^^^ x Zi, that is 

^UKh, ... A^O X ^^{hX ... hi) = iP,' + iP,'+iPi + ... + iP\. 

When the roots are all real, we have thus the product of one sum of squares 
by the product of another sum of squares (the number in each sum depend- 
ing upon the arbitrary quantity i), brought under the form of a sum of 
a constant number n of squares, which in itself is an interesting theorem. 

The expression above given for Qi leads to a remarkable relation between 

f'x 
the quotients and convergents to -7- . 

Let it be supposed, as before, that 

f^ 1 1 1 1 

fx ~ Q-yX - Q2X - QsX -'" QnOS ' 

and let the successive convergents to this continued fraction be 

A\(x) J^,{x) N,(x) Nnj^) 

D^{xy D^(x)' D,(x)' '" Dn{coy 

where the numerators and denominators are not supposed to undergo any 
reductions, but are retained in their crude forms as deduced from the law 

Ni = QiNi_,-Ni^, 

A = Qi-Di-i--Di-2- 



398 On Sturm's Method of Real Roots of an Equation. [50 

Ni (x) being 1, and A(^) being Qi {x) ; then it may be deduced from the 
published results above adverted to that 

A ('^) = fbf'""yf" UKK- K){«=-hd (00 - h,j ... {x-h,,)}. 

Hence S {? (^e, 'is, • • • ^Si-i) x (^e - ^e,) ('^e - K^ ..-{he- hg._^)} 



Z^ Z^ 

and we have therefore 



A-i(A.); 



p - ^^'-1 -^^^-3 ■^^'-5 ^^^a n /i \ 

and consequently 

«^ = %' 5=' • • • #^ 2 {( A-. (Ae)f (X - K% 

Zj Z i_4 Z (j)+] 

which is the general equation connecting the form of each quotient with 

that of the denominator to the immediately preceding unreduced convergent 

f'x 
in the expansion of "^ under the form of an improper continued fraction. 

If instead of the denominator of the unreduced convergents, the denom- 
inators of the convergents reduced to their simplest forms be employed, 
the powers of Z in the constant factor will undergo a diminution. The 
essential part of this theorem admits of being stated in general terms as 
follows : — 

" If the quotient of an algebraical function of x by its first differential 
coefficient be expressed under the form of a continued fraction whose 
successive partial quotients are linear functions of x, any one of these 
quotients may be found (to a constant factor "pres) by taking the sum of the 
products formed by multiplying each factor {x — K) of the given function by 
the square of what the denominator of the immediately antecedent conver- 
gent fraction becomes after substituting in it for x the root corresponding to 
such factor." 

P.S. Since the above was read before the British Association, the 
theory has been extended by the author to comprise the general case of 
the expansion of any two algebraical functions under the form of a continued 
fraction, and has been incorporated into the paper in the Philosophical 
Transactions above referred to. 



51. 



ON A THEOREM CONCERNING THE COMBINATION 
OF DETERMINANTS. 



[Cambridge and Dublin Mathematical Journal, viii. (1853), pp. 60 — 62.] 

Let ^A represent the line of terms ^a^, ^a^, ... ^am, 
'B „ „ „ %,%, ...-"br,,. 

Let ^A X ^jB represent 2 O^r x ^b^), where of course there are m terms 
within the symbol of summation. 

Again, let ^A represent the line ^Oj, ''aj, ... ''am, 

^B „ „ \,%, ...''bm, 

^, ^ \'A\ I >5 I ^ „ I 'ar, 'as I I %, % I 

and let „ , x „ n represent Z \ „ „ x I , „, , 
l^Al \'B\ '^ \ "-ar, 'as \ %, % I 






denoting the determinant (^a^ . ^aj — 'as . ''ar), 



there being of course ^m (m — 1) terms comprised within the sign of 
summation ; and so, in general, let 



, n being less than m. 



'A 




'B 


'A 




'B 


'A 


X 


'B 


M 




»£ 



400 On a Theorem concerning the 

/and where in general M denotes ''Oi, ^a^, ... ^a-iri\ , 

V and "^B denotes ''61, ^h^, ... %n) 

'hu„ %„ ... %, 



[51 






Now let r be any integer less than m, and let 

m{m — ] ) ... (to — r + 1) 

^ = 17277^^ ' 

and, supposing 6^, 6^, ... Or to be r numbers of the set 1, 2, ... m, let 
Gi, (z2, ... (xfi denote the fi rectangular matrices of the forms 






respectively, 



and let ^1, H^, ... H^ denote the //, rectangular matrices of the forms 

respectively. 



»rB 
Now form the determinant 

G,xH„ G,xE,, ... G,xH^ 
G,xH„ G,xH„ ... G,xH^ 

G^xH,, G^xH,, ... G^xH^ 

then, if we give r the successive values 1 , 2, 3 ... to (in which last case the 
determinant in question reduces to a single term), the values of the deter- 
minant above written will be severally in the proportions of 
K, K"", Z^(«^ii, ... K^'.K; 

that is to say, the logarithms of these several determinants will be as the 
coefficients of the binomial expansion (1 + «)™. 

When we make ?' = to, and equate the determinant corresponding to this 
value of r with that formed by making r = 1, the theorem becomes identical 
with a theorem previously given by M. Cauchy, for the Product of Kect- 
angular Matrices. 



51] Combination of Determinants. 401 

It would be tedious to set forth the demonstration of the general theorem 
in detail. Suffice it here to say that it is a direct corollary from the formula 
marked (4) in my paper in the Philosophical Magazine for April 1851, 
entitled "On the Relations between the Minor Determinants of Linearly 
Equivalent Quadratic Functions*," when that formula is particularized by 
making 

( ^nt+i ! "m+2 1 • • ■ Om+n) 

represent a determinant all whose terms are zeros except those which lie in 
one of the diagonals, these latter being all units, which comes, in fact, to 
defining that 

I J"^n = 1, and I 1""+' I = 0. 

The important theorem here referred to is made almost unintelligible 
by an unfortunate misprint of *^,„, ^6m, ^6m, '^^m, in place of '20^, ^6r, '^Qr, ''^r- 
I may here take notice of another and still more inexplicable blunder in the 
same paper, formula (S)"}", in the latter part of the equation belonging to 
which 

is written in lieu of 

«], Oj, ... ttm, ^9m+i) ''^9m+2' ••• ^flm+s "^"H-l ^«+3 • • • ^»+'nl 
Oi, a^, ... am, a.(()„+i, 0^ii.m+2> ••• '^^m+s <^»+i '^w+a • ■ • '^n+w 



[* p. 249 above.] [t See pp. 246, 251 above.] 



26 



52. 

NOTE ON THE CALCULUS OF FOEMS. 

[See pp. 363 and 411.] 

[Cambridge and Dublin Mathematical Journal, viii. (1853), pp. 62 — 64.] 

Accidental causes have prevented me from composing the additional 
sections on the Calcuhis of Forms, which 1 had destined for the present 
Number of this Journal. In the meanwhile the subject has not remained 
stationary. Among the principal recent advances may be mentioned the 
following. 

1. The discovery of Combinants ; that is to say, of concomitants to 
systems of functions remaining invariable, not only when combinations of 
the variables are substituted for the variables, but also when combinations 
of the functions are substituted for the functions ; and as a remarkable first- 
fruit of this new theory of double invariability, the representation of the 
Resultant of any three quadratic functions under the form of the square of 
a certain combinantive sextic invariant added to another combinant which 
is itself a biquadratic function of 10 cubic invariants. When the three 
quadratic functions are derived from the same cubic function, this expression 
merges in M. Aronhold's for the discriminant of the cubic. The theory 
of combinants naturally leads to the theory of invariabilitj'' for non-linear 
substitutions, and I have already made a successful advance in this new 
dii-ection. 

2. The unexpected and surprising discovery of a quadratic covariant 
to any homogeneous function in x, y of. the nth degree, containing {n — 1) 
variables cogredient with x^""^, x'^~~^y ... y""^ and possessing the property of 
indicating the number of real and imaginary roots in the given function. 
This covariant, on substituting for the (n — 1) variables the combinations of 
the powers of x, y with which they are cogredient, becomes the Hessian 
of the given function*. 

* This covariant furnishes, if we please, functions symmetrical in respect to the two ends 
of an equation for determining the number of its real and imaginary roots. The ordinary 
Sturmian functions, it is well known, have not this symmetry. As another example of the 
successful application of the new methods to subjects which have been long before the mathe- 
matical world and supposed to be exhausted, I may notice that I obtain without an effort, 
by their aid, a much more simple, practical, and complete solution of the question of the simul- 
taneous transformation of two quadratic functions, or the orthogonal transformation of one 
such function, than any previously given, even by the great masters Cauehy and Jacobi, who 
have treated this question. 



52] Note on the Calculus of Forms. 403 

3. The demonstration due to M. Hermite of a law of reciprocity connect- 
ing the degree or degrees of any function or system of functions with the 
order or orders of the invariants belonging to the system. The theorem 
itself was first propounded by me about a twelvemonth back, and com- 
municated to Messrs Cayley, Polignac, and Hermite, as serving to connect 
together certain phenomena which had presented themselves to me in the 
theory : unfortunately it appeared to contradict another law too hastily 
assumed by myself and others as probably true, and I consequently laid 
aside the consideration of this great law of reciprocality. To M. Hermite, 
therefore, belongs the honour of reviving and establishing, — to myself what- 
ever lower degree of credit may attach to suggesting and originating, — 
this theorem of numerical reciprocity, destined probably to become the 
corner-stone of the first part of our new calculus ; that part, I mean, which 
relates to the generation and affinities of forms *. 

4. I may notice that the Calculus of Forms may now with correctness 
be termed the Calculus of Invariants, by virtue of the important observation 
that every concomitant of a given form or system of forms may be regarded 
as an invariant of the given system and of an absolute form or system of 
absolute forms combined with the given form or system. As regards that 
particular branch, of the theory of invariants which relates to resultants, or, 
in other words, to the doctrine of elimination, I may here state the theorem 
alluded to in a preceding Number of the Journal, to wit that if R be the 
resultant of a system of n homogeneous functions of n variables, written 
out in their complete and most general form (so that ' ^flnition i? = 
is the condition that the equations got by making the n gi.v, ""ctions 
zero, shall be simultaneously satisfiable by one system of ratios), then the 
condition that these equations may be satisfied by t distinct systems of 
ratios between the n variables is h'-R = 0, the variation S being taken in 
respect to every constant entering into each of the n equations. 

* This theorem of numerical reciprocity promises to play as great a part in the Theory of 
Forms as Legendre's celebrated theorem of reciprocity in that of Numbers. Another demonstra- 
tion of it, which leaves nothing to be desired for beauty and simpUeity, has been since 
discovered by Mr Cayley, which ultimately rests upon that simple law (essentially although not 
on the face of it a law of reciprocity) given by Euler, which affirms that the number of modes in 
which a number admits of being partitioned is the same whether the condition imposed upon 
the mode of partitionment be that no part shall exceed a given number, or that the number of 
parts constituting any one partition shall not exceed the same number. 



26—2 



53. 



ON THE RELATION BETWEEN THE VOLUME OF A TETRA- 
HEDRON AND THE PRODUCT OF THE SIXTEEN ALGE- 
BRAICAL VALUES OF ITS SUPERFICIES. 

[Cambridge and Dublin Mathematical Journal, viir. (1853), pp. 171 — 178.] 

The area of a triangle is related (as is well known) in a very simple manner 
to the eight algebraical values of its perimeter : If we call the values of the 
squared sides of the triangle a, b, c, there will be nothing to distinguish the 
algebraical affections of sign of the simple lengths so as to entitle one to a 
preference over the other. The area of the triangle can only vanish by reason 
of the three vertices coming into a straight line; hence, according to the 
general doctrine of characteristics, we must have the Norm of \Ja, + \/6 + \/o, 
containing as a factor some root or power of the expressions for the area of 
the triangle. The Norm in question being representable as — N- where N 
is the Norm of a^ ±bi ± c^, which is of four dimensions in the elements a, b, c, 
and undecomposable into rational factors, we infer that to a numerical factor 
pres the square of the area must be identical with the Norm N, and thus, 
by a logical cnup-de-main, completely supersede all occasion for the ordinary 
geometrical demonstration given of this proposition, which in its turn, with 
certain superadded definitions, would admit of being adopted as the basis 
of an absolutely pure system of Analytical Trigonometry that should borrow 
nothing fi.-om the methods and results of sensuous or practical geometry. 
But into this speculation it is not my present purpose to enter : what I 
propose to do is to extend a similar mode of reasoning to space of three 
dimensions, and to point out a general theorem in determinants which is 
involved as a consequence in the generalization of the result of the inquiry 
when pushed forward into the regions of what may be termed Absolute or 
Universal Rational Space. 

Let F, G, H, K be the four squared areas of the faces of a tetrahedron, 
and V the volume ; then, since V only becomes zero in the case of the four 
vertices coming into the same plane, which is characterised by the equation 



53] Relation hettoeen the Volume of a Tetrahedron, etc. 405 



subsisting, we infer that N the Norm of 

must contain a power of F as a rational factor. V^ is rational and of 
three dimensions in the squared edges; the Norm above spoken of is of 
eight dimensions in the same. Consequently there is a rational factor, 
say Q, remaining, which is of five dimensions in the squared edges, and this 
factor I now proceed to determine, the other factor F- being, as is well 
known, a numerical product of the determinant 



0, 


ab-, 


ac", 


ad", 


1 


ba', 


0, 


bc\ 


bd\ 


1 


ca\ 


c6^ 


0, 


cd-, 


1 


da\ 


d¥, 


dc\ 


0, 


1 


1, 


1, 


1, 


1, 






a, b, c, d being the four angular points of the tetrahedron. See London 
and Edinburgh Philosophical Magazine, 1852. [p. 386 above.] 

The quantity Q possesses an interest of a geometrical character ; for if we 
call the radii of the eight spheres which can be inscribed in a tetrahedron 
■^i; ^2) ''s, *'4. ^B. '"e. J"?. '>'a, we evidently have riVirjrir^r^rjrf, x iV=(3F)^ Hence 

(R), the product of the eight radii in question, = — ^ = . 

Consequently Q is the quantity which characterises the fact of one or 
more of the radii of the inscribed spheres becoming infinite. For the triangle 
there exists no corresponding property ; this we know d priori, and can 
explain also analytically from the fact that if we call P the product of the 
radii of the four inscribable circles, v the Norm of the perimeter, and A 
the area, we have 

Pv = 2'A\ 



and 






which contains no denominator capable of becoming zero, so that as long 
as the sides remain finite the curvature of the inscribed circles is incapable 
of vanishing. 

To determine iV^ as a function of the edges, and then to discover by actual 

N 
division the value of ^ , would be the direct but an excessively tedious 

and almost impracticably difficult process. I have ever felt a preference 
for the a, prion method of discovering forms whose properties are known, and 
never yet have met with an instance where analysis has denied to gentle 



406 



On the Relation between the 



[53 



solicitation conclusions which she would be loth to grant to the application 
of force. The case before us offers no exception to the truth of this remark. 
Q is a function of five dimensions in terms of the squared edges : let us 
begin by finding the value of that part of Q in which at most a certain 
set of four of these edges make their appearance, and to find which con- 
sequently the other two edges may be supposed zero without affecting the 
result. We may make two distinct hypotheses concerning these two edges ; 
we may suppose that they are opposite, that is non-intersecting edges, or 
that they are contiguous, that is intersecting edges. 

To meet the first hypothesis suppose ah = 0, ce = 0. 

For convenience sake, use F, G, H, K to denote 16 times the square 
of each area, instead of the simple square of the areas. Call 



\Q{abcY = K, 16(abdy = H, 16{acdy-=G, 16 {bcdf = F. 



Then 



- if = {aby + (acy + (bey - 2 (aby (acy - 2 {aby (bey - 2 {acy (bey 
= a(f + bc*-2(acy(bcy. 

Similarly, 

-H=ad'+bd'-2(ady(bdy, 

-0 =ca'' + da^ - 2ca?da', 

-F = cb* + d¥ - 2cPd¥. 

Hence one value of i<JF + s/G + \/H + \/K will be 

V(- 1) {(ad" - he") + (hd" - ad') + (da' - ac') + (be' - bd')] = 0. 

Hence, on this first supposition, the Norm vanishes. But V does not vanish 
when ab = 0, cd = 0, for it becomes, saving a numerical factor, 



0, 


0, 


ac', 


ad'. 


1 


0, 


0, 


be', 


bd'. 


1 


ca', 


cb', 


0, 


0, 


1 


da', 


db', 


0, 


0, 


1 


1, 


1, 


1, 


1, 





that is 

(ac' . bd' — ad' . be') (cb' + ad' — ca' — bd') 

+ (be' - ac') (ca' . db' - cb' . da') 

+ (ad' — bd') (ca' . db' — cb' . da') 

= 2 (ac' . bd' — ad' . be') (ad' + be' — ac' — bd') ; 

and consequently, since iV vanishes but V' does not vanish, Q vanishes, 
showing that there is no term in Q but what contains one at least of any 



53] 



Volume of a Tetrahedron, etc. 



407 



two opposite edges as a factor; or, in other words, there is no terra in Q 
of which the product of the square of the product of all three sides of some 
one or other of the four faces does not forra a constituent part. 

Next, let us suppose ah = 0, ac = 0, then 
K^ = \Qahc' = -hc\ 
H^ = 16ab(P = - (ad- - bd% 
(? = 16acd^ = - {ad" - cdrf, 
F" = imcd' = -bc'- bd* - cd' + 2bc" . bd- + 26c^ . cd" + 2bd' . cd\ 

Four of the factors of N will be therefore 

{( {be- + cd^ - bd'') ± F}, {i {be- - cd- + bd-) ± F}, 
I denoting s/{— 1), and the product of these four factors will be 

{(6c= + cd' - bd-'Y + F"-] X {{be" - cd" + 6^^^ + F"}, 

which is equal to 

16bc*.bd".cd"; 

and similarly, the remaining part of the Norm will be 

{{2ad" - bd" - cd" + be")" + F"}x {{2ad" - bd" - cd" - be")" + F"}, 

that is 

{4a* - 4!ad" {bd" + cd" + be") + 4:hc" . bd" + 4:bd" . cd" + 4^cd" . be"} 

X {4acZ'' - iad" (bd" + cd" - be") + 4<bd" . cd"]. 
Again, since ac" = and be" = 0, V" becomes 







0, 


0, 


0, 


ad". 


1 


, 






0, 


0, 


be", 


bd", 


1 








0, 


eh", 


0, 


cd", 


1 








da", 


db", 


dc" 


0, 


1 








1, 


1, 


1, 


1, 









which is evidently equal to 














2hc" 


0, 
0, 
da" 
1, 


0, 
cb", 
db", 

1, 


ad", 
cd", 
0, 
1, 


1 
1 
1 


-be* 


0, 

da", 
1, 


ad", 
0, 
0, 


1 
1 



= 2bc" {2bc"ad" + ad^ - ad"hd" - cd-ad" + bd"cd"} - 2bc^ad" 
= 2bc" {ad'' - ad" {bd" + cd" - be") + bd" . cd"}. 



\ 



408 On the Relation between the [53 

Hence, paying no attention to any mere numerical factor, we have found that 

N 
when ac = and &c = 0, Q or p^ becomes 

hd" . b(P . cd- {ad-" - ad" (bd- + cd- + be-) + bd" . bd'' + bd- . cd- + cd^ . bc^}. 

Hence, with the exception of the terms in which five out of the six edges 
enter, the complete value of Q will be 

S (6c= . bd" . cd") {ad'' - ad"- (bd" + cd" + be") + be" . bd" + bd" . cd" + cd" . bd"], 

or more fully expressed, and still abstracting from terms containing five 
edges, 

= tbd" . bd" . cd" {(ab* + ac' + ad') - (ab" + ac" + be") {bd" + be" + cd") 

+ be" . bd" + bd" . cd" + cd" . be"}. 

It remains only to determine the value of the numerical coefficient 
affecting each of the six terms of the form 

ab" . ac" . ad" . be" . bd". 

To find this, let 

ab" = ac" = ad"=be" = bd"=cd"= 1 ; 

then evidently, since all the squared areas are equal, several of the factors 
of iV will become zero, but V" evidently does not become zero for a regular 
tetrahedron ; hence Q becomes zero : and if we call the numerical factor 
sought for X, we must have (observing that the 2 includes four parts cor- 
responding to each of the four faces) 

4 {3 - 9 + 3} + 6X, = 0, 
therefore - 12 + 6X = 0, or X = 2. 

Hence the complete value of Q is 

tab" . be" . ca" {{da* + d¥ + dc') - {da'' + db" + de") {ab"+ be" + ca") 
+ ab" . be" + be" . ca^ + ca" . ab"} 
+ 22 {ab" . be" . ed" . da" . ae") ; 

or, which is the same quantity somewhat differently and more simply 
arranged, 

Q = 2 {ab" . be" . ca") [{da' + d¥ + dc' + da" . db" + db" . dc" + de" . da") 
+ {ab" . bd" + be" . ca" + ca" . ab") - {da" + db^ + de") {ab" + be" + ear)}, 
and this quantity equated to zero expresses the conditions of a radius of an 



53] Volume of a Tetrahedron, etc. 409 

inscribed sphere becoming infinite. The direct method would have involved, 
as the first step, the formation of the Norm of a numerator consisting of 

^lF±^/G±■^H± ^]K, 

the value of which is 

2^4 _ i^sf^Q ^ 6tF'G' + 4>-%F"-GH - iOFGHK, 

and contains 4+6 + 12, that is 22 positive terms, and 12, that is 13 negative 
terms, together 35 terms, each of which might be an aggregate of 6'' or 
1296 quantities, and thus involve in all the consideration of 45360 separate 
parts, for each of the quantities F, G, H, K being a quadratic function of 
three of the squared edges, will contain six terms. It is not uninteresting to 
notice that in addition to the case already mentioned of two opposite edges 
being each zero, as ah = 0, ccl = 0, Q will also vanish for the case of ab = cd, 
bc = ad; that is for the case of two intersecting edges being each equal in 
length to the edges respectively opposite to them. This is evident from 
the fact that on the hypothesis supposed the face acb = acd and the face 

bdc = bda ; hence N= 0, and therefore, V not vanishing, ^ , that is Q, will 
vanish. 

We may moreover remark that since ab = and cd = does not make 
V vanish, the perpendicular distance of ab from cd, which, multiplied by 
ab X cd, gives six times the volumes, must on this supposition become infinite. 
When three edges lying in the same plane all vanish simultaneously, Q 
vanishes, since one edge at least in every face of the pyramid vanishes, 
and V also vanishes, as is evident from the expression for V^, when ab = 0, 
ac = 0, 6c = 0, becoming a multiple of 

0, 0, 0, ad", 1 

0, 0, 0, bd\ 1 

0, 0, 0, cd% 1 
ad% bd\ cd\ 0, 

1, 1, 1, 0, 

which is evidently zero. 

It appeared to me not unlikely, from the situation and look of Q (the 
characteristic of one of the inscribed spheres becoming infinite), that it might 
admit of being represented as a determinant, but I have not succeeded in 
throwing it under that form. I have a strong suspicion that if we take 
Q' a function corresponding to a tetrahedron a'b'c'd', in the same way as 
Q corresponds to abed, QQ', and not improbably '/(QQ'), will be found to be 



410 Relation between the Volume of a Tetrahedron, etc. [53 



(as we know from Staudt's Theorem of -^{V^ .V'^)) a rational integral 
function of the squares of the distances of the points a, h, c, d from the points 
a', V, c', d'. 

That N should divide out by V- is in itself an analytical theorem relating 
to 6 arbitrary quantities ab^, ac", ad", be'', bd", cd"^, which evidently admits 
of extension to any triangular number 10, 15, &c. of arbitrary quantities. 
Thus we may affirm, d priori, that the norm of 

^/L±^/M±^/N±^/P± ^Q, 

where (for the sake of symmetry, retaining double letters, as AB, AG, &c., 
to denote simple quantities) 



0, AB, 


AC, 


AD, 


1 




0, 


AB, 


AC, 


AE, 


1 


AB, 0, 


BC, 


BD, 


1 




AB, 


0, 


BC, 


BE, 


1 


AC, BC, 


0, 


CD, 


1 


, P = 


AC, 


BC, 


0, 


CE, 


1 


AD, BD, 


CD, 


0, 


1 




AE, 


BE, 


CE, 


0, 


1 


1, 1, 


1, 


1, 







1, 


1, 


1, 


1, 






i\^=&c., ilf=&c., i = &c., 
will contain as a factor the determinant 

0, AB, AC, AD, AE, 1 

AB, 0, BC, BD, BE, 1 

AC, BC, 0, CD, CE, 1 

AD, BD, CD, 0, DE, 1 

AE, BE, CE, DE, 0, 1 

1, 1, 1, 1, 1, 

71 (tI +1^ 

and a similar theorem may evidently be extended to the case of any — ^^ — - 
arbitrary quantities whatever. 



54 



ON THE CALCULUS OF FORMS, OTHERWISE THE THEORY 
OF INVARIANTS. 

[Continued from p. 363 above.'] 

[Cambridge and Dublin Mathematical Journal, viii. (1853), pp. 256 — 269.] 

Section VII. On Combinants. 

Reasons of convenience have induced me to depart from the plan to 
which I originally intended to adhere in the development of this theory, 
and I shall hereafter, from time to time, continue to add sections on such 
parts of the subject as may chance to be most present to my mind or most 
urgent upon my attention, without waiting for the exact place which they 
ought to occupy in a more formal treatise, and without having regard to the 
separation of the subject into the two several divisions stated at the outset 
of the first section. The present section will be devoted to a brief and 
partial exposition of the theory of Combinants*, with a view to the applica- 
tion of this theory to the solution of the problem of throwing the resultant 
of three general homogeneous quadratic functions under its most simple form, 
being analogous to that given by Aronhold in the particular case where 
the three functions are derived from the same cubic, and becoming identical 
therewith when the coefficients are accommodated to this particular supposi- 
tion"]". I shall confine myself for the present to combinants relating to 
systems of functions, all of the same degree. 

If ^1, (^2, ■■• <^r, be homogeneous functions of any number of variables, any 
invariant or other concomitant of the system which remains unchanged, not 
only for linear substitutions impressed upon the variables contained within the 
functions, but also for linear combinations impressed upon the functions them- 
selves, is what I term a Combinant. A Combinant is thus an invariant or other 
concomitant of a system in its corporate capacity (qua system), being in fact 

* Discovered by the Author of this paper in the winter of 1852. 

+ A similar method will subsequently be applied to the representation of the resultant of two 
cubic equations as a function of Combinants bearing relations to the quadratic and cubic 
invariants of a quartic function of x and y, precisely analogous to those which the Combinants 
that enter into the solution above alluded to bear to the Aronholdian invariants of a cubic 
function. 



412 On the Calculus of Forms. [54 

common to the whole family of forms designated by Xic^j + Xj^j + ... + Xr</'r. 
where X], X^, ... \r, are arbitrary constants. If the coefficients of ^j, ^2, ... <^,-, 
be supposed to be written out in r lines (the coefficients of corresponding 
terms occupying the same place in each line), so as to form a rectangular 
matrix, any combinantive invariant will be a function of the determinants 
corresponding to the several squares of r- terms each that can be formed out 
of such matrix, or, as they may be termed, the full determinants belonging 
to such rectangular matrix. If we call any such combinant K, then, over 
and above the ordinary partial differential equations which belong to it in its 
character of an invariant, it will be necessary and sufficient, in order to 
establish its combinantive character, that K shall be subject to satisfy (r — 1) 
pairs of equations of the form 

da do dc J 

d , d d \ „ ^ 

da do dc J 

where a,b,c...; a',b',c'..., are respectively lines in the matrix above 
referred to. 

So any combinantive concomitant will be a function of the full deter- 
minants of the matrix formed by the coefficients of the given system of forms 
and of the variables, and will be subject to satisfy the additional differential 
equations just above written. 

It will readily be understood furthermore, that an invariant or other 
concomitant may be combinantive in respect to a certain number of forms 
of a system, and not in respect of other forms therein ; or more generally, 
may be combinantive in respect of each, separately considered, of a series of 
groups into which a given system may be considered to be subdivided, 
without being so in respect of the several groups taken collectively. 

In the fourth section of my memoir [p. 429 below] on a "Theory of the 
Conjugate Properties of two rational integral Algebraical Functions," recently 
presented to the Royal Society of London, the case actually arises of an 
invariant of a system of three functions, which is combinantive in respect 
only to two of them. 

For greater simplicity, let the attention for the present be kept fixed 
upon combinants which are such in respect of a single group of functions, 
all of the same degree in the variables. (It will of course have been 
perceived that when the system is made up of several groups, there would 
be nothing gained by limiting the groups to be all of the same degree 
inter se; it is sufficient that all of the same group be of the same degree 
per se.) 



54] On the Calculus of Forms. 413 

All such combinants will admit of an obvious and immediate classification. 
Let us suppose that a combinant is proposed which is in its lowest terms, 
that is to say, incapable of being expressed as a rational integral algebraical 
function of combinants of an inferior order. Such a combinant may, notwith- 
standing this, admit of being decomposed into non-combinantive invariants 
of inferior dimensions to its own, and in such event will be termed a complex 
combinant ; or it may be indecomposable after this method, in which event 
it will be termed a simple combinant. It will presently be shown, that the 
resultant of a system of three quadratic functions is made up of a complex 
combinant of twelve dimensions, and of the square of a simple combinant 
of six dimensions, expressible as a biquadratic function of ten non-com- 
binantive invariants, each of three dimensions in the coefficients. There 
is an obvious mode of generating complex combinants ; according to which 
they admit of being viewed as invariants of invariants. Supposing 
(^1, ^2, ... ^r, to be the functions of the given system, Xi(^i + 7i2<^24-... +\r^r 
may conveniently be termed the conjunctive of the system: if now one or 
more invariants or other concomitants be taken of this conjunctive, there 
results a derivative function or system of functions of the quantities 
Xi, X-i, ... Xr, in which every term affecting any power or combination of 
powers of the X series is necessarily an invariant or concomitant of the 
given system. If now an invariant or other concomitant be taken of the 
new system in respect to Xj, Xa, ... Xy, (the original variables (supposing them 
to enter) being treated as constants), this secondarily derived invariant will 
be itself an Invariant, or at all events a Concomitant in respect of the 
original system, and being unaffected by linear substitutions impressed upon 
the X system, is by definition a combinant of such system. A similar 
method will obviously apply if the original system be made up of various 
groups; each group will give rise to a conjunctive, and one or more con- 
comitants being taken of this system of conjunctives and treated as in the 
case first supposed, (the only difference being, that there will on the present 
supposition be several unrelated systems instead of a single system of new 
variables, that is, several X systems instead of one only) the result, when all 
the X systems have been invariantized out (that is, made to disappear by any 
process for forming invariants), will be a combinant in respect to each of the 
groups, severally considered, of the given system of functions. 

Here let it be permitted to me to make a momentary digression, in order 
to be enabled to avoid for the future the inconvenience of using the phrase 
" invariant or other concomitant," and so to be enabled at one and the same 
time to simplify the language and to give a more complete unity to the 
matter of the theory, by showing how every concomitant may in fact be 
viewed as a simple invariant, so that the calculus of forms may hereafter 
admit of being cited, as I propose to cite it, under the name of the Theory 
of Invariants. 



414 On the Calculus of Forms. [54 

Thus, to begin with the case of simple contragredience and cogredience, 
if ?> '7. ^•■- ^i'6 contragredient to x,y,z..., any form containing ^, ■?;, ^..., 
which is concomitantive to a given form or system of forms S, which contains 
x,y,z..., may be regarded as concomitantive to the system 8' , made up of 
;Si and the superadded absolute form ^x + rjy + ^z-\- ..., say &; where f, t;, f... 
are treated no longer as variables, but as constants. In like manner every 
system of variables contragredient to x, y, z ..., or to any other system of 
variables in 8, will give rise to a superadded form analogous to ^, the totality 
of which may be termed 8-^, ; and thus the various sj^stems ^,77, ^ ... will no 
longer exist as variables in the derived form, but purely as constants. Again, 
if 8 contain any system of variables (/>, i\r, S-, &c., contragredient to x, y, z, &c., 
the system of variables u, v, w, &c., cogredient with x, y, z, &c., may be 
considered as constants belonging to the superadded form ^u + -^v + '^w ... ; 
but if 8 do not contain any system contragredient to x, y, z, &c., then 
u, V, w, &c. may be treated as constants belonging to the superadded system 
of forms XV — yu, yw — zv, zu — xw, kic; and so in general any concomitant 
containing any sets of variables in simple relation, whether of cogredience 
or contragredience, with any of the sets in the given system 8, may in all 
cases be treated as an invariant of the system 8', made up of 8 and a 
certain superadded system 81, all the forms contained in which are ab- 
solute, by which I mean, that they contain no literal coefficient. The same 
conclusion may be extended to the case of concomitants containing sets of 
variables in compound relation with the sets in the given system of forms 8. 
Thus, suppose Mj, 1*2, ... Wn, to be in compound relation of cogredience with 
a;"""\ x^~^y, x'^~^y^, ... y"'~'-; u^, u.,, ... m„, may be regarded as constants 
belonging to the superadded form 

«i2/"~^ — (w — 1) u^y^-'x + ^ (?i — 1) (n — 2) u^y^-^x' + . . . + u^x^-^; 

say n. And thus universally we are enabled to affirm, that a concomitant 
of whatever nature to a given system of forms, may be reduced to the form 
of an invariant of a system made up of the given system and a certain other 
superadded system of absolute forms : without, therefore, abandoning the use 
of the terms concomitant, cogredience, contragredience, &c., which for many 
purposes are highly convenient and save much circumlocution, we may 
regard every concomitant as a disguised invariant, and under the name of 
the Theory of Invariants comprise the totality of the theory of Concomitance. 
I have already had occasion to make use of the superadded form fl in 
discussing the theory of the Bezoutiant (a quadratic form concomitant to 
two functions of the same degree in x, y, which plays a most important part 
in the theory of the relations of their real roots), in the memoir for the Royal 
Society previously adverted to. 

I now return to the question of applying the theory of combinants to 
the decomposition of the resultant of three general quadratic functions of 



54] On the Calculus of Forms. 415 

X, y, z. It will of course be apparent that every resultant of any system of n 
functions of the same degree of a single set of n variables is a combinantive 
invariant of the system. This is an immediate and simple corollary to the 
theorem given by me in this Journal, in May, 1851. Accordingly, in pro- 
ceeding to analyse the composition of the resultant of three quadratic 
functions, I may, besides impressing linear combinations upon the variables, 
impress linear combinations upon the functions themselves, in any way most 
conducive to simplicity and facility of expression and calculation ; and 
whatever relations shall be proved to exist between the resultant and other 
combinants for such specific representation, must be universal, and hold good 
for the functions in their most general form. 

(1) The system, by means of linear substitutions impressed upon the 
variables which enter into the functions, may be made to assume the form 

x^ + y^ + z^, 

ax^ + hy"^ + cz"^, 

Ix- + my'' + nz- + 2pyz + 2qzx + 2rxy. 

(2) By means of linear combinations of the functions themselves the 
system may evidently be made to take the form 

(c — a)x" + (c — b) y-, 

(a -b)y'' + (a- c) z-, 

hy- + 2,pyz + 2qzx + 2rxy ; 

and finally, by taking suitable multipliers of x, y, z in lieu of x, y, z, it may 
be made to become 

P («' - y% 

a- {f - z% 
y'^ + ^fyz + 'igzx + 2hxy. 

We have thus reduced the number of constants in the system from 
eighteen to five ; and as it will readily be seen that in any combinant of the 
system in its reduced form p and o- can only enter as factors of the simple 
quantity, (pa-y, for all purposes of comparison of the combinants of the 
system of like dimensions with one another, p and cr might admit of being 
treated as being each unity, and accordingly, practically speaking, we have 
only to deal with three in place of eighteen constants, a marvellous simplifi- 
cation, and which makes it obvious, a priori, or at least affords a presumption 
almost amounting to and capable of being reduced to certainty, that the 
number of fundamental combinants of the system, of which all the rest must 
be explicit rational functions, will be exactly four in number ; which, for the 
canonical form hereinbefore written, on making p and a each unity, will 
correspond to 

1, f+g' + h^ fy + g^^ + h'f, fgh. 



416 



On the Calculus of Forms. 



[54 



and will be of the 3rd, 6th, 12th, and 9th degrees respectively. The reason 
why the squares of /, g, h, instead of the simple terms /, ff, h, appear in the 
2nd and 3rd of these forms is, because, on changing x into —oo,y into —y, 
or z into — z, two of the quantities /, g, h will change their sign, but the 
forms representing the invariants of even degrees ought to remain absolutely 
unaltered for such transformations. I shall in the course of the present 
section set forth the methods for obtaining these four combinants, which, 
although of the regularly ascending dimensions 3, 6, 9, 12, belong obviously 
to two different groups, the one of three dimensions forming a class in itself, 
and the natural order of the three others being that denoted by the sequence 
6, 12, and 9, and not that which would be denoted by the sequence 6, 9, 12, 
the combinant of the ninth degree being properly to be regarded as in some 
sort an accidentally rational square root of a combinant of 18 dimensions. 

Let now /^ (^ — y") = ^> 

a{y^-z^)= W, 
y^ + 2/2/^ + Igzx + Ihxy = V. 
The resultant will be found by making 
« = ± y, 
■2 =±2/, 
when a; = + 2/] 

^ = + 2/) 
x = ^■y\ 
z = -y\ 
x = -y\ 
z = + y\ 
x = -y\ 
z = -y] 

Hence the resultant It 
= p'(7' (1 + 2/+ 2g + 2h) (1 - 2/- 2g + 2h) (1 + 2/- 2g - 2h) (1 - 2/+ 2g - 2h) 
= {pay {(1 + 2hy - 4 (/+ gy] {(1 - 2/0'- - 4 (/- gy] 
= {pay {(1 + ^h? - 4/ - ^g^y - (4A - Sfgy] 
= {pay [1 _ 8 (/ + 5r= + /i=) + 16 {{f + g^ + ¥)-2 {g^lf + h^f +f-g^] + 64/^A]. 

Let now K = W -}- iJ,V + v'W, 

K being what I term a linear conjunctive of U, V, W. The invariant of K, 
in respect to x, y, z, will be the determinant 

pX, hfi, g/M 

hfi, fi — pX + av, ffi 
91^' M - o"" 



V = {l+2f+2g+2h)y', 
V={l-2f-2g + 2h)f, 
V={l + 2f-2g-2h)y% 
V={l-2f+2g-2h)f. 



54] 



On the Calculus of Forms. 



417 



that is 

= {Ifgh - g') fi'+a (h^ - g'') /m'v -pif- g") /x'X - p<rfjXv + /sWi/ - pa^Xv'' ; 
or, multiplying by 6, we may write 

Ix,y,zK = (ydXfiv + Sbsf^^v + Sbifi'X + Sa^X-v + SciXv- + ftjAtS 
where d = — per, h = IS/p'/i — 6g^, 

h = -2p{f'-g% h=^a{h?-g% 
a, = p-a; Ci = — 2pa^, 

the notation being accommodated to that employed by Mr Salmon in The 
Higher Plane Curves, \, fi, v in IK being correspondent to x, y, z in 
Mr Salmon's form. If now we employ Mr Salmon's expression for the S 
(the biquadratic Aronholdian of IK), observing that 
ttg = 0, C2 = 0, Oi = 0, C3 = 0, 
we have the complex combinant 

S\,^,^Ix,y,zK = d'^-id' (61 Ci + tTa&a) + da^b^Cj, - a^cAh + bi^cj' + a^-bi 
(I _ 8 (/+/,= - 2^0 + 4 {12fgh - 6g^) \ 

^- 1 6 (/ - f) (h? -g') + lQ if' - gj + Q^' - fP 
= p'<7'{l-8 if' + g' + h') + 16 if' + g'+ U - hY - 9'P -f'h') + ^gh]. 
Hence, calling the resultant R, we have 
-2R + 4^^,^,, h^y,, K = \-S{f^ + g'' + h') + 16 {f + g' + h') 

+ 32 (/Y + g'h' + h'p) = {1 - 4 (/= + ^= + li?)Y = P'. 
Let O be taken the polar reciprocal to the conjunctive 
-\U+fiV+vW; 
and for greater simplicity, as we know, a priori, from the fimdamental 
definition of a combinant, which (save as to a factor) must remain unaltered 
by any linear modification impressed upon the functions to which it apper- 
tains, that p and o- can enter factorially only in any combinant, let p and cr 
be each taken equal to unity in performing the intermediary operations. 



^p'o-" 



Then 



12 = 



-\ 


hf^, 


gfj-, 


1 


h/j,^ 


X + fM + V, 


U, 


f] 


an-' 


M 


-~ V, 


K 


I 


V, 


t 






+ -rf (- \v + ^^V) 
_ J + f ^ (X== + X/Ci + Xv + /i>=) 
~l-2^r(/V + %/^0 

+ 2r? {g (pX + fMv) + (g -fh) p?] 

- 2^97 {hp,v +fgp?) 



27 



418 



On the Calculus of Forms. 



[54 



Upon n, which is a quadratic function in respect of each of the two 
unrelated systems ^,17, f ; X, jm, v, and also in respect of the coeflScients in 
{TJ, V, W), we may operate with the coramutantive symbol 



d 


d 


d\ 


dV 


dt] ' 


d^ 


d 


d 


d 


dV 


df]' 


d^ 
d_^ 


d 


d 


d\' 


dfji' 


dv 


d 


d 


d 


d\' 


d/j,' 


dv 



which, for facility of reference, I shall term 8E. 

Considering the first line as statiooary, we shall obtain, for the value of 
8E (Q,), 216 commutantives, which may be expressed under the following 
forms : 

drj ' d^ 

A A. 

dr]' d^ 

* ill 



d/u? 



dr 

A 
dr 

I- 



A 
dr 

dr 

d- d d 
d\^ ' dfjb dv' 

A A 
dr dr]' 

A 
dr 

d^d_ 

dX dv' 
A 

dr 

A 
dr 

d^d_ 

dX dfi' 



A 
drj' 

A 

drj ' 



A 

drj ' 

A. 
djj?' 

A 

dri ' 

A 

drj' 

d 



A 

d^ 

A 

d^ 

A i 

dfi d 

A 
dK 

A 

dK 

d d 
dX d) 

d 
dK 

A 
dt 

d d- 



g 



y 



dX dfi ' d. 



^ 



54] 



071 the Calculus of Forms. 



419 



d 


d d 


dr 


dr;' d^ 


d 


d d 


rfr 


dt)' d^ 


'd d 


d d d d' 


dX da' 


du dv ' dv d\ 





In this expression the first lines may be considered stationary, the 
second lines are subject to the usual process of commutation, which makes 
three of the six permutations positive and three negative ; and the third 
or bracketed lines are subject to the simple process which makes all the 
permutations of the same sign. In the three middle groups two of the 
terms in the final line are always identical ; it will therefore be more 
convenient to introduce the multiplier 2, and then to consider each such line 
to represent the three distinct permutations, taken singly. 

Let now 

d^ d^ 1 



d^ 
dr_ 
d d 



drj' 

dri d^ 
8 [d^d^' df)^ 



AL\n- 

d^r 



IT.' x::f.Ki = (")'. 



AA 

dt] rff 
d^d\;,\ 



^y\^ = ^^)" 



fi = (n)" 



8 \dr^d'^' di]d%' d^. 

d d d d d d\^^ _/o^ 
^^' l^dX' ^rf|J "-Wi- 



And let 



Til ^ ^1 

\dX- ' dy? ' dv^^ 

l_dX° ' djji. dv 

\A ^ ^ 
\_dX dv ' dfj,^ 



= L, 



dfji ' 

d_ 
dX 



. dv} ' 
, dvj ' 

rd^d^ ± ± dn^ J,,, 

\_dX dfj, ' dX dfi' dv-J 

r d d d d d d~\ _ J- 
[dX dfj, ' d/j, dv' dv dxj 

Then, attending to the convention just previously explained, we shall have 



E (D.) ={L- 2L' - 2i" - 2L'" + 2 A) 

X ((fl) - 2 (O)' - 2 (f))" - 2 (O)"' + 2 (H),}, 



27—2 



420 On the Calculus of Forms. [54 

a symbolical product, any term in which such as L'Q," will mean 

if— ^ i- — ^\ 

\_d\^ ' dfidv' d/x, dv] 

jd _d d^ d^d^ 1^ 

d^d^' dr' d^d^ 1 

and a similar interpretation must be extended to each of the 25 partial 
products ; we have then 

L(n) = 8g\ -2L'{a) = 0, -2L"'(n) = 0, 

-2i"(n) = -v. 2Zi(n) = -2, 
-2Z(n)'=o, -2L(n)"' = o, 

4i'(^)' = 0. 4X"(fl)"' = 0, 
4iL"(D,y = 0, 4>L"'{D.)"' = 0, 
4X"'(n)' = 8/S 4Z' (fiy" = 8/i,^ 
-2i(fl)" = 0, 4Z'(I1)"=0, 4Z"(n)" = 0, 4Z"'(fi)" = 0, 
-4i:i(Il)' = 0, -4Zi(O)"' = 0, 
- 4Zi (H)" = 4c^= ; " 
and, finally, the five terms comprised in 

2i(nx, ..., 4ij(nx, 

each = 0. All the above equations can be easily verified by direct inspection, 
it being observed that 8 (O) represents 

v"- + Xv + fiv +f'fJ'-, — Xi/ + g-fx,', X^ + X/ji + Xv + h^f/?, 
that 8 (fl)' represents 

v' + iJiv + Xv^ f-jjr, -fX/j, - %/^^ -/X/t - hfffj,'', 
that 8 (fl)" represents 

-Xv+g'-fj,\ g(ijX + /j.v) + (g-fh)/M', g {fjX + ij,v) + {g - fh) fi\ 
that 8 (12)'" represents 

X= + yaX + i/A, + /i>=, — hiJLv - /^'yu,-, - hfxv —fgfJ?, 
and that (fl)i represents 

-fXjx - hgii\ g {/jX + p.v) + {g -fh) /x^ - hfiv -fg/J?. 

We have thus 

E (fl) = 8g"- - 4/ - 2 + 8/^ + 8h' + 4/ 

= 2{if + 4>g- + ih'-l}. 



Hence 



SR = iS,^^,J,,,,,K-l{EnY. (A) 



54] On the Calculus of Forms. 421 

If we restore to U, V, W their general values, and make 
U = ad^ + hy' + cz- + ^fyz + Igzx + IKxy, 
V= aV + by + c'z' + 2f'yz + 2g'zx + 2h'xy, 
W = a"x- + V'y'^ + c V + 2f"yz + 2g"zx + 2K'xy, 
and construct the cubic function 

S- = {ax + a'y H- a"z) {hx + h'y + h"z) {ex + c'y + c"z) 

— {ax + a'y + a"z) {fx +/'y -\-f"zy — (bx + b'y + b"z) {gx + g'y + g"zy 

— {ex + c'y + c"^') {hx + A'y + A"^)^ 

+ 2 O +/ V +/"^) {gx + ^r'y + ^r"^) {hx + A'y + A"^), 

that is 

2 (a6c - af - bg^ - ch' + 2fgh) a? 

+ 2 [a'bc + ab'c + a6c' - {a'p + 2aj5r') - {b'g"- + 2bgg') - {c'h^ + 2cAA') 

+ 2f'gh+2fg'h + 2fgh'}x''y 

+ \a'b"c + a'bc" + a"b'c + a"bc' + ab'c" + ab"c' — 2a' ff" — 2af'f" — 2a!' fj' 

— 2b' gg" — 2bg'g" — 2b" gg — 2c'hh" — 2ch'h" — 2c"AA' 

+ 2/"sr'A + 2/'sr"A + 2fg'h" + 2f"gh' + 2f"gh' + 2fg"h'} xyz, 

S\,^,vlx,y,tl^ in the preceding equation becomes simply the Aronholdian 
S to ^, which may be calculated by Mr Salmon's formula previously quoted. 

D, may be taken equal to the determinant 

ax + a'y + a"z, hx + h'y + h"z, gx + g'y + g"z, f 

hx4-h'y + h"z, bx + b'y + b"z, fx +f'y +f"z, tj 

gx + g'y + g"z, fx+f'y+f'z, cx + c'y+c"z, ? 

e V, r. 

And the cubic commutant of this, obtained by affecting it with the com- 
mutantive operator, 

d d d -\ 

dx' dy' dz 

d d d 
dx' dy' dz 

d d d 
d^' di)' df 

d d d 
d^' drj' d^-l 



422 On the Calculus of Forms. [54 

will give 48^ (fl) if each of the four lines of the operator undergoes permuta- 
tion, or SE{D,), if one of the four lines is kept stationary. Thus it falls 
within the limits of practical possibility to calculate explicity, by the formula. 
(A), the value of the resultant. I give to the (S of ^ the appellation of the 
Hebrew letter \^{shin), and to the commutant of H the appellation of the 
Hebrew letter b {teth). These letters are chosen with design ; for I shall 
presently show that when the three given quadratic functions are the 
differential derivatives of the same cubic function i/r, the b becomes the 
Aronholdian T to the cubic function, or, as we may write it, T-y^r, and the 
JJ' becomes the Aronholdian S of the Hessian thereto, that is SHyjr. 

Thus for the first time the true inward constitution of the resultant of 
three quadratics is brought to light. The methods anteriorly given by me,^ 
and the one subsequently added by M. Hesse for finding this resultant, 
adverted to in Section II., lead, it is true, to the construction of the form, 
but throw no light upon the essential mode of its composition. 



55. 



TH^OREME SUE LES LIMITES DES RACINES REELLES DES 
l^QUATIONS ALG^BRIQUES. 

[JS^ouvelles Annates de Mathematiques, xii. (1853), pp. 286 — 287.] 

SoiT f(x) = 

une Equation alg^brique de degr6 n, et supposons qu'en operant sur f(x) 
etf (x) comme dans le theoreme de M. Sturm, on obtienne les n quotients 

OiX + bi, UifK + b^, asX+bs-.-anX + bn', 

il faut remarquer seulement qu'on obtient le Ji'^™^ quotient, a„a; + &«, en 
divisant I'avant-dernier residu par le dernier rdsidu. 

Formons la sdrie de 2n quantites 

+ 2-6i ±2-b . +2-63 ±2-bn . 

il n'y a aucune racine de lequation 

f(x) = 

entre la plus grande de ces quantites et + co , ni entre la plus petite de ces 
quantitds et — 00 * 

* Prochainemeiit, une demonstration de ee theortoe generalise, [p. 424 below.] 



56. 



NOUVELLE METHODE POUR TROUVER UNE LIMITE SUPERI- 
EURE ET UNE LIMITE INFERIEURE DES RACINES 
R^ELLES D'UNE l^QUATION ALG^BRIQUE QUELCONQUE. 

[NouveUes Annales de MatMmatiques, xii. (1853), pp. 329 — 336.] 

1. Lemme. Soient 

une suite de quantit^s positives, assujetties a cette loi 

Oi = fli, G2 = fl2-\ 1 Gs = fl3-\ ...Ci = fiiA ...Gr = Hr, 

ou les fi sont des quantites positives quelconques. 
Si, dans la fraction continue 

i 1 i _L 1 

qi + q2 + q3+ •■• + gr-i + g-r ' 

(les quantites ^i, ^2 ... ^tant des quantity positives ou negatives), on a les 
in6galites 

[qi] > Ci, fc] > C„ [q,] >C,... [qr-,] > Gr-u [?r] > Gr 

(les crochets indiquent la racine carrde positive du carre de la quantity que 
ces crochets renferment), le denominateur de la fraction continue aura m^me 
signe que le produit qiq^qs... qr-iqr- 



Dimonstration. Posons 



qi = »ii, 



1 


= m^, 




= mi, 


1 

qr+ 


= mr; 



A 



56] Nouvelle Methode, etc. 425 

il est aisd de verifier que les d^nominateurs successifs de la fraction continue 
sont 

nil, miTHi, mim^mi, , mimjms ... m^-imr", 

mi a meme signe que qi-. 

11 1 lllrnlrnl^ 

- = — , r-i< -> — < - . [?2] > — . [?2] > — . etc. ; 

done ^2 a meme signe que m,, et aussi mi^Tij est de meme signe que qiq^: 
1 1 1 r n 1 

done 5'3 a meme signe que ms; ainsi inim^ins est de meme signe que qiq^qs, 
et, en continuant, on parvient k demontrer que mim^ms ... m^-imr, c'est-a-dire 
le d^nominateur de la fraction continue, est de meme signe que le produit 
qiq,q3-.-qr-iqr. 

2. THjfioRfeME. Si f{x) est une fonction algdbrique entiere de degre n, 
et si Von prend arhitrairement une autre <f> (x) algdbrique et entiere, et d'un 

degre moindre que n, et qu'on developpe la fraction y , en fraction continue , 

f(x) ~Xi + X,+ ...+Xr-i + X/ 

oil, Xi, X^... Xr sont des fonctions rationnelles de x, et si Von forme V equation 

(0) (Xi' - Gi^) (X/ - Ci) . . . (xv-i - av-a) (Xr' - G/) = 0, 

la racine r^elle superieure de cette Equation sera plus grande, et la racine rdelle 
inferieure de cette equation sera moindre qu'aucune des racines rdelles de 
V6quation 

/(^) = 0; 

et si toutes les racines de Vdquation (6) sont imaginaires, Vdquation 

/(^) = 0, 

aura aussi toutes ses ramies imaginaires. 

Demonstration. Tous les quotients de la fraction continue qui suivent le 
premier quotient, savoir : Xs, X, ... X^, sont en gdndral des fonctions lin^aires 
de X, et Xi sera aussi line'aire, si j> (x) est de degr^ n—1; les cas particuliers 
ne changent pas la marche de la demonstration ; mais il faut remarquer 
que lorsque f(x) et <^ (x) ont des racines communes, le dernier quotient aura 

la forme ^ , [x] e'tant I'avant-dernier terme, et alors, dans I'dquation (9), 

au lieu de X^" — C/, on ^crit simplement X/. 




Nouvelle Metliode, etc. [56 

Soient L la plus grande racine et A la plus petite racine de I'^quation {6) ; 
alors aucun facteur de {6) ne peut devenir nul pour des valeurs de a; comprises ; 
entre + oo et Z, et entre A et — oo ; done on aura toujours 

m>c„ 



[X,-\>Gr. 

Or f{x) est evidemment egal au d^nominateur de la fraction continue 
multipli^ par un facteur constant. Done, en vertu du lemme, le ddnominateur 
de la fraction continue est de meme signe que le produit X^X^X^ ... X^-^Xr 
pour les valeurs de x comprises entre + oo et i, et entre A et — oo ; mais dans 
ces intervalles la fonction gdn^rale Xi n'etant pas comprise entre + 0^ et — Gi 
ne peut devenir nulle, et, par consequent, ne peut changer de signe ; done le 
denominateur de la fraction continue conserve le meme signe pour touts 
valeur de x renfermee entre ces intervalles, et de meme/'(a;) ; L est done une 
limite superieure et A une liraite infdrieure des racines de 1 equation 

/(*) = o. 

Le nombre des racines reelles de I'equation {6) est dvidemment pair, zero 
compris ; dans ce dernier cas, c'est-a-dire (6) n'ayant aucune racine reelle, 
/(«) ne changera done pas de signe pour des valeurs de x comprises entre 
+ 00 et — 00 ; autrement toutes les racines de /(«) = sont imaginaires. Le 
theoreme est done completement demontre. 

3. Si (/) (x) est de degre n — 1, la fraction continue renferme en general 
(sauf les cas oh. qtielques-uns des coefficients deviennent nuls), comme il 
a ete dit plus haut, n quotients lineaires de la forme 

aiX — bi, cux — h^.-.an-iX — hn-i, anX — bn] 

done, d'apres le theoreme, la plus grande et la plus petite des 2w quantites 

6i±Ci h±G, 6„_i + Gn-i bn ± Cn 

sont respectivement une limite superieure et une limite inferieure des 
racines de I'dquation 

f{x) = Q. 
Si Ton prend (r = n) 

on vient au theoreme enonce [p. 423]. 



56J Nouvelle Methode, etc. 427 

4. Lors meme que les quotients X^, X^, etc., ne sont pas lin^aires, on 
n'aura pourtant jamais a re'soudre que des equations du premier degr^. En 
effet, soient les 2r equations de degrd quelconque 

Zi-C, = 0, Z,-C', = 0...Z,-O, = 0, 

Xi + (7i = 0, X^+G^ = O...Xr+Gr = 0. 

II suffit de trouver une quantite I sup^rieure aux racines de ces equations, 
et une quantite X inferieure k ces m^mes racines, Z et A. seront des limites 
pour lequation 

Si done une de ces equations est de degre ^ > 1, on applique a cette equation 
le precede ci-dessus, en choisissant une fonction {oo) de degr^ p — 1, et, en 
agissant ainsi, on arrivera par une sorte de trituration a n'avoir a traiter que 
des Equations du premier degr^. 

5. On a 

plus la valeur de fii est petite, et plus on aura de chances a resserrer les 
limites dans les deux fractions -^-= — - ; par centre, on aura un desavantage sous 

ce rapport dans les deux fractions suivantes '"'"' ~ — ^— ; car C^+i = /x;+i H — ; 

plus /i; diminue, et plus Cj+i augmente. Cet inconvenient n'a pas lieu pour 

la derniere fraction ; on pent done prendre fjun = et 0^ = . 

A'ji— 1 

6. II est a remarquer que tons les raisonnements precedents subsistent 
en renversant la suite des fi et I'^crivant ainsi : 

11 1 

7. II y a lieu a des recherches int^ressantes sur la forme a donner h, 
j>{x), et sur les valeurs a donner aux quantites yx pour obtenir les limites 
les plus resserrees, et je crois etre parvenu a demontrer que la forme la plus 
avantageuse est/' («), precisement la forme que M. Sturm a adoptee. 

8. Dans la reduction en fraction continue de .; , , nous n'avons con- 

sidere que des quotients binomes ; mais on pent pousser les divisions plus 
loin et obtenir des quantity de la forme 

T c d I 

«« + &+-+ - + ... + — ; 



% 



n 



428 Nouvelle Methode, etc. [56 

le reste correspondant sera de la forme 

aV+i + &V + cV-i +...+ — . 

En operant ainsi, le nombre de termes dans chaque reste ira en diminuant, 
comme dans le procdde ordinaire, et le dernier reste sera de la forme Caf-, 
(i, etant un entier positif ou negatif, et le dernier quotient de la forme 
PxP + QxP~'^, p etant|un entier positif ou negatif; nommant les quotients 
ainsi obtenus q^, q^ ... q^, on voit aisement qu'on aura 

ou M est une constante, i un nombre entier positif ou negatif dont la valeur 
depend de la maniere dont on a opere dans les divisions successives, et D est 
le denominateur de la fraction continue 

111 1 1 

21 + 92 + qs + ■■■ + qr-l + qr' 
Done, si Ton dorit, comme ci-dessus, 

X = (q,' - Cr) (q,' - Q) . . . (q/ - Or') = 0, 

nommant i et A les racines extremes de cette equation, si z^ro n'est pas 
compris entre + oo et i, ni entre A et — oo , la demonstration donne'e ci- 
dessus subsiste encore pour le cas general. Et lors meme que zdro est 
compris entre ces limites, Z et A restent tout de meme les limites pour les 
racines, abstraction faite de la racine zero. 



m. 



ON A THEOKY OF THE SYZYGETIC* RELATIONS OF TWO 
RATIONAL INTEGRAL FUNCTIONS, COMPRISING AN 
APPLICATION TO THE THEORY OF STURM'S FUNCTIONS, 
AND THAT OF THE GREATEST ALGEBRAICAL COMMON 
MEASURE. 

[Philosophical Transactions of the Royal Society of London, CXLIII. (1853), 
Part III., pp. 407—548.] 

Introduction. 

" How charming is divine philosophy ! 
Not harsh and crabbed as dull fools suppose, 
But musical as is Apollo's lute, 
And a perpetual feast of nectar'd sweets, 
Where no crude surfeit reigns ! " — Comds. 

In the first section of the ensuing memoir, which is divided into five 
sections, I consider the nature and properties of the residues which result 
from the ordinary process of successive division (such as is employed for the 
purpose of finding the greatest common measure) applied to f{x) and ^ (*•), 
two perfectly independent rational integral functions of x. Every such 
residue, as will be evident from considering the mode in which it arises, 
is a syzygetic function of the two given functions ; that is to say, each of the 
given functions being multiplied by an appropriate other function of a given 
degree in x, the sum of the two products will express a corresponding residue. 
These multipliers, in fact, are the numerators and denominators to the 

successive convergents to ^ expressed under the form of a continued frac- 
jx 

tion. If now we proceed a priori by means of the given conditions as to 



* Conjugate would imply something very different from Syzygetic, namely, a theory of the 
Invariantive properties of a system of two algebraical functions. 



430 On a Theory of the Syzygetic Relations [57 

the degree in x of the multipliers and of any residue, to determine such 
residue, we find, as shown in Art. 2, that there are as many homogeneous 
equations to be solved as there are constants to be determined ; accordingly, 
with the exception of one arbitrary factor which enters into the solution, 
the problem is definite ; and if it be further agreed that the quantities 
entering into the solution shall be of the lowest possible dimensions in 
respect of the coefEcients of / and ^, and also of the lowest numerical 
denomination, then the problem (save as to the algebraical sign oi plus or 
minus) becomes absolutely determinate, and we can assign the numbers 
of the dimensions for the respective residues and syzygetic multipliers. 
The residues given by the method of successive division are easily seen not 
to be of these lowest dimensions ; accordingly there must enter into each 
of them a certain unnecessary factor, which, however, as it cannot be 
properly called irrelevant, I distinguish by the name of the Allotrious 
Factor. The successive residues, when divested of these allotrious factors, 
I term the Simplified Residues, and in Arts. 3 and 4 I express the 
allotrious factor of each residue in terms of the leading coefficients of the 
preceding simplified residues of f and (f>. In Art. 5 I proceed to determine 
by a direct method these simplified residues in terms of the coefficients 
of / and <p. Beginning with the case where f and <^ are of the same 
dimensions {m) in x, I observe that we may deduce, from _/ and (p, m linearly 
independent functions of x each of the degree (m — 1) in x, all of them 
syzygetic functions of / and (/> (vanishing when these two simultaneously 
vanish), and with coefficients which are made up of terms, each of which 
is the product of one coefficient of/ and one coefficient of <p. These, in fact, 
are the very same m functions as are employed in the method which goes 
by the name of Bezout's abridged method to obtain the resultant to (that is, 
the result of the elimination of x performed upon) / and <p. As these derived 
functions are of frequent occurrence, I find it necessary to give them a name, 
and I term them the m Bezoutics or Bezoutian Primaries ; from these m 
primaries m Bezoutian secondaries may be deduced by eliminating linearly 
between them in the order in which they are generated, — first, the highest 
power of X between two, then the two highest powers of x between three, 
and finally, all the powers of x between them all: along with the system 
thus formed it is necessary to include the first Bezoutian primary, and to 
consider it accordingly as being also the first Bezoutian secondary ; the last 
Bezoutian secondary is a constant identical with the Resultant of/ and <p. 
When the m times m coefficients of the Bezoutian primaries are conceived 
as separated from the powers of x and arranged in a square, I term such 
square the Bezoutic square. This square, as shown in Art. 7, is sym- 
metrical about one of its diagonals, and corresponds therefore (as every 
symmetrical matrix must do) to a homogeneous quadratic function of m 
variables of which it expresses the determinant. This quadratic function. 



57] of two Algebraical Functions. 431 

which plays a great part in the last section and in the theory of real roots, 
I term the Bezoutiant; it may be regarded as a species of generating 
function. Returning to the Bezoutic system, I prove that the Bezoutian 
secondaries are identical in form with the successive simplified residues. 
In Art. 6 I extend these results to the case of / and <f> being of different 
dimensions in x. In Art. 7 I give a mechanical rule for the construction 
of the Bezoutic square. In Art. 8 I show how the theory of f{x) and (^ {x), 
where the latter is of an inferior degree to /, may be brought under the 
operation of the rule applicable to two functions of the same degree at the 
expense of the introduction of a known and very simple factor, which in fact 
will be a constant power of the leading coefficient in /(»). In Art. 9 I give 
another method of obtaining directly the simplified residues in all cases. 
In Art. 10 I present the process of successive division under its most general 
aspect. In Arts. 11 and 12 I demonstrate the identity of the algebraical 
sign of the Bezoutian secondaries with that of the simplified residues, 

generated by a process corresponding to the development of ^ under the 

form of an improper continued fraction (where the negative sign takes the 
place of the positive sign which connects the several terms of an ordinary 
continued fraction). As the simplified residue is obtained by driving out 
an allotrious factor, the signs of the former will of course be governed by the 
signs accorded by previous convention to the latter ; the convention made is, 
that the allotrious factors shall be taken with a sign which renders them 
always essentially positive when the coefficients of the given functions are 
real. I close the section with remarking the relation of the syzygetic 
factors and the residues to the convergents of the continued fraction which 

expresses -^ , and of the continued fraction which is formed by reversing 
jx 

the order of the quotients in the first named fraction. 

In the second section I proceed to express the residues and syzygetic 
multipliers in terras of the roots and factors of the given functions ; the 
method becoming as it may be said endoscopic instead of being exoscopic*, 
as in the first section. I begin in Arts. 14 and 15 with obtaining in this 

* These words admit of an extensive and important application in analysis. Thus the 
methods for resolving an equation (or to speak more accurately, for making one equation depend 
upon another of a simpler form) furnished by Tschirnhausen and Mr Jerrard (although not so 
presented by the latter) are essentially exoscopic ; on the other hand, the methods of Lagrange 
and Abel for effecting similar objects are endoscopic. So again, the memoir of Jacobi, " De 
Eliminatione," hereinafter referred to, takes the exoscopic, and the valuable " Nota ad Elimina- 
tionem pertinens " of Professor Bichelot in Crelle's Journal, the endoscopic view of the subject. 
In the present memoir (in which the two trains of thought arising out of these distinct views are 
brought into mutual relation) the subject is treated (chiefly but not exclusively) under its 
endoscopic aspect in the second, third and fourth sections, and exoscopically in the first and last 
sections. 



432 On a Theory of the Syzygetic Relations [57 

way, under the form of a sum or double sum of terms involving factors 
and roots of f and (^, and certain arbitrary functions of the roots in each 
term, a general representative, or to speak more precisely, a group of general 
representatives for a conjunctive of any given degree in x to /and 0, that is, 
a rational integral function of x, which is the sum of the products of / and 
(^ multiplied respectively by rational integral functions of x, so as to vanish 
of necessity when /and ^ simultaneously vanish. This variety of representa- 
tives refers not merely to the appearance of arbitrary functions, but to an 
essential and precedent difference of representation quite irrespective of such 
arbitrariness. 

In Arts. 16, 17, 18, 19, 20, 21, I show how the arbitrary form of function 
entering into the several terms of any one (at pleasure) of the formulae that 
represent a conjunctive of any given degree may be assigned, so as to make 
such conjunctive identical in form with a simplified residue of the same 
degree. The form of arbitrary function so assigned, it may be noticed, 
is a fractional function of the roots, so that the expression becomes a sum 
or double sum of fractions. I first prove in Arts. 16, 17 that such sum is 
essentially integral, and I determine the weight of its leading coefficient in 
respect of the roots of / and ^ (this weight being measured by the number 
of roots of / and ^ conjointly, which appear in any term of such coefficient). 
Now in the succeeding articles I revert to the Bezoutic system of the first 
section, and beginning with the supposition of m and n being equal, I demon- 
strate that the most general form of a conjunctive of any degree in x will be 
a linear function of the Bezoutics, from which it is easy to deduce that the 
simplified residues of any given degree in x are the conjunctives whose 
weight in respect of the roots is a minimum; so that all conjunctives having 
that weight must be identical (to a numerical factor pres), and any integral 
form of less weight apparently representing a conjunctive must be nugatory, 
every term vanishing identically. These results are then extended to the 
case of two functions of unlike degrees. The conclusion is, that the weight 
of the forms assumed in Arts. 16 and 17 being equal to the minimum weight, 
they must (unless they were to vanish, which is easily disproved) represent 
the simplified residues, or which is the same thing, the Bezoutian secondaries. 

We thus obtain for each simplified residue a number of essentially 
distinct forms of representation, but all of which must be identical to a 
numerical factor pres, a result which leads to remarkable algebraical 
theorems. 

The number of these different formulse depends upon the degree of the 
residue ; there being only one for the last or constant residue, two for the 
last but one, three for the last but two, and so on. The formulee continue to 
have a meaning when their degree in x exceeds that of / or (^ ; but then, 
as although always representing conjunctives, they no longer represent 



57] of two Algebraical Functions. 433 

residues, this identity no longer continues to subsist. In Arts. 22, 23, 24, 25, 
I enter into some developments connected with the general formulge in 
question ; these, it may be observed, are all expressed by means of fractions 
containing in the numerator and denominator products of differences; the 
differences in the numerator products being taken between groups of roots 
of f and groups of roots of ; and in the denominator between roots of f 
inter se and roots of (f> inter se. A great enlargement is thus opened out to 
the ordinary theory of partial fractions. 

In Art. 26 I find the numerical ratios between the different formulse 
which represent (to a numerical factor p7~es) the same simplified residue, 
and in Arts. 27 and 28 I determine the relations of algebraical sign of these 
formulse to the simplified residues or Bezoutian secondaries. In Art. 29 
I determine the syzygetic multipliers corresponding to any given residue 
in terms of the factors and roots of the given functions ; but the expressions 
for these, which are closely analogous to those for the residues, cease to be 
polymorphic. They are obtained separately from the syzygetic equation, 
and it is worthy of notice, that to obtain the one we use the first of the 
polymorphic expressions for the residue, and to obtain the other the opj)osite 
extremity of the polymorphic scale. In the subsequent articles of this 
section, by aid of certain general properties of continued fractions, I establish 
a theorem of reciprocity between the series of residues and either series of 
syzygetic multipliers. 

Section III. is devoted to a determination of the values of the preceding 
formuloa for the residues and multipliers in the case applicable to M. Sturm's 
theorem, where (px becomes the differential derivative of fx. It becomes 
of importance to express the formulae for this case in terms of their roots 
and factors oifx alone, without the use of the roots and factors of f'x, which 
will of course be functions of the former. 

By selecting a proper form out of the polymorphic scale, the fractional 
terms of the series for each residue in this case become separately integral, 
and we obtain my well-known formula for the simplified residues (Sturm's 
reduced auxiliary functions) in terms of the factors and the squared differ- 
ences of partial groups of roots. This is shown in Art. 35. In Art. 36 the 
multiplier oi f'x in the syzygetic equation is expressed by formulas of equal 
simplicity, and in a certain sense complementary to the former. This 
method, however, does not apply to obtaining expressions for the multiplier 
of fx in the same equation in terms of the roots and factors of fx ; for the 
separate fractions whose sum represents any one of these factors, it will 
be found, do not admit of being expressed as integral functions of the roots 
and factors. To obviate this difficulty I look to the syzygetic equation itself, 
which contains five quantities, namely, the given function, its first differential 
derivative, the residue of a given degree, and the two multipliers, all of 
s. 28 



434 On a Theory of the Syzygetie Relations [57 

which, except the multiplier of /a;, are known, or have been previously deter- 
mined as rational integral functions of the roots and factors of fx. I use 
this equation itself for determining the fifth quantity, the multiplier in 
question. To perform the general operations by a direct method required 
for this would be impossible ; the difficulty is got over by finding, by means 
of the syzygetie equation, the particular form that the result must assume 
when certain relations of equality spring up between the roots oi fx; and 
then, by aid of these particular determinations, the general form is demon- 
stratively inferred. 

This investigation extends over Arts. 38, 39, 40, 41, 42, 43. It turns 
out that the expressions for the multipliers of fx are of much greater 
complexity than for the multipliers oif'x or for the residues. Any such 
multiplier consists of a sum of parts, each of which, as in the case of the 
residues and the factors oi fx, is affected with a factor consisting of the 
squared differences of a group of roots ; but the other factor, instead of being 
simply (as for the residues and factors before mentioned) a product of certain 
factors oi fx, consists of the sum of a series of products of sums of powers 
by products of combinations of factors of fx, each of which series is affected 
with the curious anomaly of its last term becoming augmented in a certain 
numerical ratio beyond what it should be in order to be conformable to the 
regular flow of the preceding terms in the series*. 

The fourth section opens with the establishment of two propositions 
concerning quadratic functions which are made use of in the sequel. Art. 44 
contains the proof of a law which, although of extreme simplicity, I do not 
remember to have seen, and with which I have not found that analysts are 
familiar : I mean the law of the constancy of signs (as regards the number 
of positive and negative signs) in any sum of positive and negative squares 
into which a given quadratic function admits of being transformed by 
substituting for the variables linear functions of the variables with real 
coefficients. This constant number of positive signs which attaches to 
a quadratic function under all its transformations, which is a transcen- 
dental function of the coefficients invariable for real substitutions, may be 
termed conveniently its inertia, until a better word be found. This inertia 
it is shown in Art. 45, by aid of a theorem identical with one formerly given 
by M. Cauchy, is measured by the number of combinations of sign in the 
series of determinants of which the first is the complete determinant of the 
function, the second, the determinant when one variable is made zero, the 
next, the determinant when another variable as well as the first is made 
zero, and so on, until all the variables are exhausted, and the determinant 

* The syzygetie multipliers are identical witli the numerators and denominators (expressed in 
their simplest form) of the successive convergents to the continued fraction which expresses — ^ . 



57] of tioo Algebraical Functions. 435 

becomes positive unity. In Art. 46 I give some curious and interesting 
expressions for the residues and syzygetic multipliers, under the form of 
determinants, communicated to me by M. Hermite ; and in Art. 47 I show 
how, by the aid of the generating function which M. Hermite employs, 
and of the law of inertia stated at the opening of the section, an instan- 
taneous demonstration may be given of the applicability of my formulae for 
M. Sturm's functions for discovering the number of real roots of fa, without 
any reference to the rule of common measure; and moreover, that these 
formulas may be indefinitely varied, and give the generating function, out 
of which they may be evolved, in its most general form. Had the law of 
inertia been familiar to mathematicians, this constructive and instantaneous 
method of finding formulae for determining the number of real roots within 
prescribed limits would, in all probability, have been discovered long ago, 
as an obvious consequence of such law. I then proceed in Arts. 48 and 49, 
to inquire as to the nature of the indications afforded by the successive 
simplified residues to two general functions / and ^ ; and I find that the 
succession of signs of these residues serves to determine the number of roots 
of y or (^ comprised between given limits, after all pairs of roots of either 
function contained within the given limits and not separated by roots of the 
other function have been removed, and the operation, if necessary, repeated 
toties quoties until no two roots of either function are left unseparated by 
roots of the other ; or in other words, until every root finally retained in one 
function is followed by a root of the other, or else by one of the assigned 
limits. The system of roots comprised between given limits thus reduced 
I call the effective scale of intercalations ; such a scale may begin with a root 

of the numerator or of the denominator of -^ ; and upon this and the 

fa 
relative magnitudes of the greatest root of ^x and fa it will depend whether 
in the series of residues (among which fa and ^x are for this purpose to be 
counted) changes will be lost or gained as x passes from positive infinity to 
negative infinity. In Art. 50 I observe that the theory of real roots of a 
single function given by M. Sturm's theorem is a corollary to this theory 
of the intercalations of real roots of two functions, depending upon the well- 
known law, that odd groups of the limiting function f'x lie between every 
two consecutive real roots of fa. In Art. 51 I verify the law of reciprocity, 
already stated to exist between the residues of fa and (/>«, by an cb posteriori 
method founded on the theory of intercalations. In Arts. 52, 53, 54, I obtain 
a remarkable rule, founded upon the process of common measure, for finding 
a superior and inferior limit in an infinite variety of ways to the roots of any 
given function. This method stands in a singular relation of contrast to 
those previously known. All previous methods (including those derived 
through Newton's Rule) proceed upon the idea of treating the function 
whose roots are to be limited as made up of the sum of parts, each of which 

28—2 



436 On a Theory of the Syzygetic Relations [57 

retains a constant sign for all values of the variable external to the quantities 
which are to be shown to limit the roots. My method, on the other hand, 
proceeds upon the idea of treating the function as the product oi factors 
retaining a constant sign for such values of the variable. In Art. 55, the 
concluding article of the fourth section, I pijint out a conceivable mode in 
which the theory of intercalations may be extended to S3'stems of three or 
more functions. 

In Section V. Arts. 56, 57, I show how the total number of effective 
intercalations between the roots of two functions of the same degree is given 
by the inertia of that quadratic form which we agreed to term the Bezoutiant 
to / and (^ ; and in the following article (58) the result is extended to 
embrace the case contemplated in M. Sturm's theorem; that is to say, 
I show, that on replacing the function of « by a homogeneous function of 
X and y, the Bezoutiant to the two functions, which are respectively the 
differential derivatives of / with respect to x and with respect to y, will 
serve to determine by its form or inertia the total number of real roots and 
of equal roots in f{x). The subject is pursued in the following Arts. 59, 60. 
The concluding portion of this section is devoted to a consideration of the 
properties of the Bezoutiant under a purely morphological point of view ; 
for this purpose / and are treated as homogeneous functions of two 
variables x, y, instead of being regarded as functions of x alone. In Arts. 
61, 62, 63, it is proved that the Bezoutiant is an invariantive function of the 
functions from which it is derived ; and in Art. 64 the important remark is 
added, that it is an invariant of that particular class to which I have given 
the name of Combinants, which have the property of remaining unaltered, 
not only for linear transformations of the variables, but also for linear 
combinations of the functions containing the variables, possessing thus a 
character of double invariability. In Arts. 65, 66, I consider the relation 
of the Bezoutiant to the differential determinant, so called by Jacobi, but 
which for greater brevity I call the Jacobian. On proper substitutions 
being made in the Bezoutiant for the m variables which it contains (m 
being the degree in x, y of _/ and 0), the Bezoutiant becomes identical with 
the Jacobian to /and ^ ; but as it is afterwards shown, this is not a property 
peculiar to the Bezoutiant ; in fact there exists a whole family of quadratic 
forms of m variables, lineo-linear (like the Bezoutiant) in respect of the 
coefficients in f and (/>, all of which enjoy the same property. The number 
of individuals of such family must evidently be infinite, because any linear 
combination of any two of them must possess a similar property; I have 
discovered, however, that the number of independent forms of this kind 
is limited, being equal to the number of odd integers not greater than the 
degree of the two functions / and ^. In Arts. 67 and 68, I give the means 
of constructing the scale of forms, which I term the constituent or funda- 



57] of two Algebraical Functions. 437 

mental scale, of which all others of the kind are merely numerico-linear 
combinations. This scale does not directly include the Bezoutiant within it, 
and it becomes an object of interest to determine the numbers which connect 
the Bezoutiant with the fundamental forms; this calculation I have carried 
on (in Arts. 69, 70, 71) from m = 1 to m = & inclusive, and added an easy 
method of continuing indefinitely. In this method the numbers in the 
linear equation corresponding to any value of m are determined successively, 
and each made subject to a verification before the next is determined, there 
being always pairs of equations which ought to bring out the same result for 
each coefficient. 

In the next and concluding Art. 72, 1 remark upon the different directions 
in which a generalization may be sought of the subject-matter of the ideas 
involved in M. Sturm's theorem, and of which the most promising is, in my 
opinion, that which leads through the theory of intercalations. Some of the 
theorems given by me in this paper have been enunciated by me many 
years ago, but the demonstrations have not been published, nor have they 
ever before been put together and embodied in that compact and organic 
order in which they are arranged in this memoir, — the fruit of much thought 
and patient toil, which I have now the honour of presenting to the Royal 
Society. 

P.S. In a supplemental part to the third section I have given expressions 
in terms of the roots of ^x and fx for the quotients which arise in developing 

^- under the form of a continued fraction, and some remarkable properties 
jx 

concerning these quotients. In a supplemental part to the fourth section 

I have given an extended theory of my new method of finding limits to the 

real roots of any algebraical equation. This method, so extended, possesses a 

marked feature of distinction from all preceding methods used for the same 

purpose, inasmuch as it admits in every case of the limits being brought up 

into actual coincidence with the extreme roots, whereas in other methods a 

wide and arbitrary interval is in general necessarily left between the roots 

and the limits. 



438 On a Theory of the Syzygetic Relations [57 



Section I. 

On the complete and simjylified residues generated in the process of developing 
under the form of a continued fraction, an ordinary rational algebraical 
fraction. 

Art. 1. Let P and Q be two rational integral functions of a;, and suppose 
that the process of continued successive division leads to the equations 

P - M,Q +R, = 0\ 

Il,-MA + E, = Of, (1) 



so that 

which is what I propose to call an improper continued fraction, differing from 
a proper only in the circumstance of the successive terms being connected 
by negative instead of positive signs. 

ilfo, Ml, M2, &c., i2i, Pa; Psj &c. are, of course, functions of x: the latter 

we may agree to call the 1st, 2nd, 3rd, &c. residues (in order to avoid the use 

of the longer term " residues with the signs changed ") ; and by way of 

distinction from what they become when certain factors are rejected, we may 

call Pi, P2, P3, &c. the complete residues. Each such complete residue 

No 
will in general be of the form — ^- , N, and D, being integral functions of the 

coefficients only of P and Q, but p^ an integral function of these coefficients, 

N 
and of a; ; p^ may then be termed the tth simplified residue, and yr-' the tth 

allotrious factor. Suppose P to be of m and Q of n dimensions in x, and 
m — n = e, the process of continued division may be so conducted, that all the 
residues may contain only integer powers of .x; and we may upon this 
supposition make Ma of e dimensions, and M^, M^, M^, &c. each of one 
dimension only in x; so that P,, Pj, P3, ... will be respectively of (n — 1), 
(n — 2), {n — 3), &c. dimensions in x. 



57] of two Algebraical Functions. 439 

P and Q are supposed to be perfectly unrelated, and each the most general 
function that can be formed of the same degree. From (1) we obtain 

R, = M,q-P 

= {M,M,-1)Q-M,P ^ (3) 

R, = {M,MM + M, + M,)Q- {M,M, -1)P 

&c. = &c. 

and in general we shall have 

R. = Q.Q + P.P, (4) 

where it is evident that Q^ will be of e + {i — 1), and P^ of (t — 1) dimensions 
in w. 

Art. 2. Hence it follows that the ratios Pi : Q^ : R, may be ascertained 
by the direct application of the method of indeterminate coefficients, for Q^ 
will contain e + 1, and P, will contain t disposable constants, making e + 2t 
disposable constants in all. Again, Q^Q and P^P will each rise to the degree 
n + e + i—l ina;; but their sum R^ is to be only of n — c dimensions in x. 
Hence we have to make (n+ e + i—l) — {n — i), that is e + 2t — 1 quantities 
(which are linear in respect to the given coefficients in P and Q, as well as in 
respect to the new disposable constants in Pi and Q,) all vanish, that is to 
say, there will be e + 2t.— l linear homogeneous equations to be satisfied by 
means of e + 2i disposable quantities : the ratios of these latter are, therefore, 
determinate, so that we may write 

P. = Xi(Pi)^ 

i?i = Xi(Pi)j 

and when (PJ, (Qi), (R,) are taken prime to one another, it is obvious that 
(i2.) will be in all of e + 2t dimensions in the given coefficients, that is of t in 
respect of the coefficients of P, and of e + 1 in respect of those of Q ; \i will 
correspond to what I have previously called the allotrious factor; being in 
fact foreign to the value of Pi as determined by means of the equation (4), 
and arising only from the particular method employed to obtain it through 
the medium of the system (1): it becomes a matter of some interest and 
importance to determine the values of this allotrious factor for different 
values of t*. 

* These are identical with what I termed quotients of succession in the London and Edinburgh 
Philosophical Magazine (December, 1839) [p. 43 above] ; but by an easily explicable error of 
inadvertence, the quantities Q^, Q„, &c. therein set out are not as they are therein stated to be. 



440 On a Theory of the Syzygetic Relations [57 

Art. 3. This may be done by the following method, which is extremely 
simple, and would admit of a considerable extension in its applications, were 
it not beside my immediate purpose to digress from the objects set out in 
the title to the memoir, by entering upon an investigation of the special or 
singular cases which may arise in the process of forming the continued 
fraction, when one or more of the leading coefficients in any of the residues 
vanish ; such an inquiry would require a more general character to be 
imparted to the values of the quotients and residues than I shall for my 
present purposes care to suppose. 

Let us begin with supposing e = 1, and write 

/= aa;» + 6«"-i + c«"-= + &c. ) , " 

^ = ax"-"- + ^x"--^ + 7*"-^ + &c. j ' 

Let -dr be the first residue of "^ , and w of -r- , and therefore of -^ , so that 

f 

to is the second residue of"—. 
<P 

Let to = X, (to), 6) being entirely integer, and A, a function of the coefficients 

N 
in / and (f>. If we make A, = yr , iV and D being integer functions, D will 

evidently be L^, where L denotes the first coefficient in the simplified residue 
a--v|r, and is evidently of two dimensions in a, yS, &c., and of one in a, h, &c. ; 
Dui is therefore of 2 x 2 + 1, that is five dimensions in a, /3, &c., and of two 
dimensions in a, b, &c. ; but w (by virtue of what has been observed of the 
equations in system (5)) is of three dimensions in a, /3, &c., and of two in 
a, b, &c. Hence N is of two dimensions in a, /3, &c., and of none in a, b, Sac. 
This enables us at once to perceive that N = a". 

For ^jr is of the form /— (px + g) </>, ~| 

and CO is of the form <p — {px + q')'^j 

the quotients of succession or allotrious factors themselves, but the ratios of each such to the 
one preceding, if in the series ; so that — 

Qi is \ 



&c 

This error is corrected by my distinguished friend M. Sturm {Liouville's Journal, t. viii. 1842, 
Sur un th^or^me d'Algebre de M. Sylvester), who appears, however, to have overlooked that I 
was obviously well acquainted with the existence aud nature of these factors, and their essential 
character, of being perfect squares in the case contemplated in his memoir and my own. 
MM. Borohardt, Terquem, and other writers, in quoting my formula for M. Sturm's auxiliary 
functions, have thus been led into the error of alluding to them as completed by M. Sturm. 



57] 



of two Algebraical Functions. 



441 



but iV" = makes a vanish, and therefore, upon this supposition, / and </> 
would appear to have a common algebraical factor -il^, that is to say, N 
vanishing would appear to imply that the resultant of / and ^ must vanish, 
so that N would appear to be contained as a factor in this general resultant, 
which latter is, however, clearly indecomposable into factors — a seeming 
paradox — the solution of which must be sought for in the fact, that the 
equation iV = is incompatible with the existence of the usual equations (7) 
connecting/, ^, i/r and to : but this failure of the existence of the equations 
(7) (bearing in mind that N has been shown to be a function only of the 
set of coefficients a, (3, &c.), can only happen by reason of a. vanishing when- 
ever N vanishes ; a must therefore be a root of N, or which is the same thing, 
N a power of a and hence N = a?. 

The same result may be obtained a, posteriori by actually performing the 
successive divisions ; if the coefficients of any dividend be a, b, c, d, &c., and 
of the divisor a, y8, 7, S, &c., the first remainder, fonning the second divisor, 
will be easily seen to have for its coefficients — 



a, 


b, 


d 


1 


0, 


«) 


13 


c? 


a. 


/3, 


B 





a, h, c 
0, a, /3 



Hence the coefficients in the next remainder (making 

will be each of the form of the compound determinant, — 
a. /S, 7 



a, 


b, e 


0, 


a. 13 


a, 


/3, e 



&c. 



a, b, c 
0, a, 13 
a, 13, 7 



= m) 



a, b, c 




0, a, /3 


, 


a, ^, 7 





a, 


b, 


c 




a, 


h, 


d 


0, 


a, 


/3 


, 


0, 


a, 


7 


a, 


/3, 


7 




a, 


13, 


S 


a, 


b, 


d 




a, 


b, 


e 


0, 


«, 


7 


, 


0, 


a, 


B 


M. 


/3, 


S 




«. 


13, 


e 



The compound determinant above written will be the first coefficient 
in the remainder under consideration; the subsequent coefficients will be 
represented by writing f, (f>; g, 7, &c., respectively in lieu of e, e. Omitting 

the common multiplier — , the determinant above written is equal to 



442 



On a Theory of the Syzygetic Relations 



[57 



a, 


h, c 




0, 


«, /8 


X 


a, 


/3, 7 





a, 


^', 


e 


0, 


a, 


S 


«> 


A 


6 



a, 


6, 


d 




0, 


a, 


7 


X 


«> 


/3, 


8 





a, 


h, 


cZ 


0, 


«. 


7 


a, 


/S, 


S 



/3, 



h, d 

CL, y 



a, A 7 



■ d^a- + cya- + aa (/SS — 7-)), 



The last written pair of terms are together equal to 
a, b, c 
0, a, j3 
a. /5, 7 

■which is of the form a'A — a-/S^ (/3S — 7=) « ; and the sum of the first written 
pair is of the form a-B + (tt/S- a^B — ay/3 ay^) a. Hence the entire deter- 
minant is of the form a- (A + B), showing that a^ will enter as a factor into 
this and every subsequent coefficient in the second remainder, as previously 
demonstrated above. 

It may, moreover, be noticed, that this remainder, when a^ has been 
expelled, will for general values of the coefficients be numerically as well 
as literally in its lowest terms, as evinced by the fact that there exist terms 
(for example aa-ye) having + 1 for their numerical part. The same explicit 
method might be applied to show, that if the first divisor were e degrees 
instead of being only one degree in x lower than the first dividend, a^+^ 
would be contained in every term of the second residue : the difficulty, 
however, of the proof by this method augments with the value of e ; but the 
same result springs as an immediate consequence from the method first 
given, which remains good mutatis mutandis for the general case, as may 
easily be verified by the reader. Applying now this result to the functions 
P and Q, supposed to be of the respective degrees n and n — e in x, and calling 
the coefficients of the leading terms in the successive simplified residues 
«i, ^2) «3i &c., and denoting by a the leading coefficient in Q, and as before denot- 
ing the successive allotrious factors by Xj, Xo, &c., it will readily be seen that 



^■%-i-.. 




^3'^2=~> ^4^3 


= — „, &c., 


that is 








-%4, 


^2 = y 

ttl- 


^^-a^+W ^' 


a'^+'a^'^ 


"" a.^ai ' 


and in general 










^2«l+l = ^ 








X,™. = «"+' — 


a^W ■■■ o?^-2 





(8) 



Hi-Os-as 



57] of two Algehraical Functions. 443 

Art. 4. Strictly speaking, we have not yet fully demonstrated that the 
complete allotrious factors are represented by the values above given for X, 
but only that these latter are contained as factors in the allotrious factors ; 
we must further prove that there exist no other such factors. This may be 
shown as follows : it is obvious from the nature of the process that the 
complete residues will always remain of one dimension in respect of the given 
coefficients, that is, first of one dimension in the set a, h, c, &c., and of zero 
dimensions in a, ^, y, &c. ; then conversely, of one dimension in a, /3, 7, &c., 
and of zero dimensions in a, h, c, &c., and so on, the residues being evidently 
required to conform in their dimensions to those of the first dividend and the 
first divisor alternately. These coefficients then are always of unit dimensions 
in respect to the given coefficients ; whereas it has been shown (Art. 2) that 
the simplified residues in respect to these coefficients are successively of the 
dimensions 2 + e, 4 + e, 6 + e, &c. 

Let the complete residue corresponding to X;,„ be MX^m'^^m, that is 
M — 7 7. ^.■■- —. a-m. 



or say ML; in passing from a^ to Ojg+i the dimensions rise 2 units for all 
values of q except zero, and when ^' = the dimensions increase per saltum 
frofn 1 to 2 + e ; hence the total dimensions of L in the joint coefficients 
will be 

{(e + 1 ) - 2 (e + 2)] - 4 (m - 1) + 4m + e = 1, 

and therefore M is of zero dimensions, and X^^, is the complete allotrious 
factor. In like manner if the complete residue corresponding to X^im+x he 
ilfXjm+iMsin+i, that is 

^ 1 «!= ai Cl?m,-i 

or say ML, the dimensions of L will be 

- (e + 1) - 4»i + {e + 2 (2m + 1)), that is, 1, 

and hence, as in the preceding case, M is of zero dimensions, and Xj^+i is the 
complete allotrious factor. 

Art. 5. I proceed to show how the simplified residues may be most 
conveniently obtained by a direct process, identical with that which comes 
into operation in applying to the two given functions of x the method 
familiai'ly known under the name of Bezout's abridged method of elimination. 
Let us call the two given functions Xf and F, and commence with the case 
where TJ and V are of equal dimensions (n) in x. The simplified tth residue 
will then be a function oin — l dimensions in «, and of t dimensions in respect 
of each given set of coefficients, and may be taken equal to YJJ-\- CTiF, where 
Y^ and XJ^ are each of (t — 1) dimensions in x. 



444 On a Theory of the Syzygetic Relations [57 

Let 

U = a„x'^ + aifl;""^ + a«x^~"- + ... + a„, 

y = h^x^ + &ia.'"-i + &,a;"-2 + ...+&„; 

we may write in general, m being taken any positive integer not exceed- 
ing n, 

U = (aoX"'+ Oiaj^-H . . . + «„,,) «""'" + (am+i x''-"'-^ + am+,a;»-"'-=^ + . . . + a„), 

F=(6oa;'" + &ia,''"-i-|- ... +6„,)a;"-"'-l- (6,„+ia;"-"'-i + 6,„+2a;''-™-^ + ... + &„). 

Hence 

(So*'" + hix'"'-'- + ...+ &,„) [/■ - (ao«™ + fli «'""' + . . . + a„,,) F 

= ,„/^i«"-i + ^^2*"-= + m^3^"-' + . . . + mKn, (9) 

where if we use (r, s) to denote arbs — asK for all values of r and s, we have 

^K, = (0, m + 1), ,„A% = (0, m + 2) + (1, m + 1), 

„,K, = (0, m + 3) + (1, m + 2) + (2, vi + 1), 

and in general ^^i = S (?", s), the values of r and s admissible within the sign 
of summation being subject to the two conditions, one the equality r+s=m+i, 
the other the inequality r less than i. By giving to vi all the different values 
from to m — 1 in succession, and calling 

boX">- + bix"^-' +...+bm, cux"" + aia-™-i + . . . + a^ 

respectively Q,„ and P,„, we have 



Q,U - P,V= K,x^-^+ K,x^-^+...+ Kn 
QM - P„V= JiV-'+ .K,x^--+...+ 2Kn 



(10) 



The right-hand members of these 7i equations I shall henceforth term the 
Bezoutians to U and F. 

The determinant formed by arranging in a square the n sets of coefficients 
of the n Bezoutians, and which I shall term the Bezoutian matrix, gives, as 
is well known, the Resultant (meaning thereby the Result in its simplest 
form of eliminating the variables out) of U and F. 

Eliminating dialytically, first a;""^ between the first and second, then a.'"~^ 
and a;"~- between the first, second and third, and so on, and finally, all the 
powers of x between the first, second, third, ... ?ith of these Bezoutians, and 
repeating the first of them, we obtain a derived set of n equations, the 
right-hand members of which I shall term the secondary Bezoutians to U 
and F, this secondary system of equations being 



57] of two Algebraical Functions. 445 



yai) 



(,K, Q„ - K,Q,) U - (,K, Po-K,P,)V= L,x-' + L,x--' +... + L,_, 
{UK, ,K, - ,K, ,K,) Q, + („K, K, - K, ,K,) Q, + {K, ,K, - ,K, ,K,) Q.} U 
- {(J{, ,K, - ,K, ,K,) P„ + {,K, K, - K, „K,) P, + {K, ,K, - ,K, ,K,) P,] V 

&c. = &c. 

And we can now already without difficulty establish the important proposition, 

that the successive simplified residues to -^, expanded under the form of an 

improper continued fraction, abstracting from the algebraical sign (the 
correctness of which also will be established subsequently), will be repre- 
sented by the n successive Secondary Bezoutians to the system U, V. 

For if we write the system of equations (1 1) under the general form 

'^,U-H,V= A.x"--'- + A*'"-'-' + &c., 

the degree of ^^ and H, in x will be that of Q,_-, and Pi_i, that is t — 1 ; and 
the dimensions of A,, B„ &c., in respect of each set of coefficients is evidently 
L\ consequently, by virtue of Art. 2, A^x^~^ + B^x^~^ + &c., which is the 
fth Bezoutian, will (saving at least a numerical factor of a magnitude and 
algebraical sign to be determined, but which, when proper conventions are 
made, will be subsequently proved to be +1) represent the ith simplified 

residue to ^*, as was to be shown. 

Art. 6. More generally, suppose U and V to be respectively of n + e and 
11 dimensions in x. 

Let U=a„ «"+<= + ai aj^+^-i + a^ x^+'-- + &c. 

F =&„«"+&! a;"-i-|-&c. 



JJ = (a„«^+'» + aia;«+'«-i + &c. + ae+m) «""'"■ + (a,+m+i«"-™-^ + &c. + a„+e), 
V = (6„«™ + b^x-""-' + . . . + &™) «"-™ + (6m+i«""™"" + &c. + bn), 
we obtain the equation 

Q™ U- Pe+rJ^= m^i«"+^-^ + mK^OO^+'-' + &C. + ^n+c, (12) 

* F is supposed to be taken as the first divisor, and the term residue is used, as hitherto in 
this paper, throughout in the sense appertaining to the expansion conducted, so as to lead to an 
improper continued fraction, in that sense, in fact, in which it would, more strictly speaking, be 
entitled to the appellation of excess rather than that of residue. 



446 On a Theory of the Syzygetic Relations [57 

where 

Qm = (io*™ + • • • + &m), Pe+m = (ao«^+™ + . . . + a,+^) ; 
,„7l 1 = «.„ 6m+i ; m-^2 = tto &m+2 + Oi &m+i ; ... ,„-S'e = Cto &m+e + «! &m+e-i + &C. + «(,&„ ; 

By giving to m every integer value from to (n — 1) inclusive, we thus 
obtain n equations of the form of (12), each of the degree « + e — 1 in x, and 
of one dimension in regard to each set of coefficients. 

In addition to these equations we have the e equations of the form 

a;MF= h^x^+i"- + ftiaj'^+f-i + &c. + &„«'', (13) 

in which fj, may be made to assume every value from to (e — 1) inclusive, 
and the right-hand side of the equation for all such values of fj, will remain 
of a degree in x not exceeding n + e — \, the degree of the equations of the 
system above described. There will thus be e equations in which only the 
(b) set of coefficients appear, and n equations containing in every term one 
coefficient out of each of the two sets. 

The total number of equations is of course n + e. Between the e 
equations of the second system (13) and the r occurring first in order of the 
first system (12), we may eliminate dialytically the e + r — 1 highest powers 
of X, and there will thus arise an equation of the form 

e^_JJ-We+r-xy=Lx-^'^ + L'x^-''-^ + kc.+{L), (14) 

where 6r-\ and coe+r-i are respectively of the degrees r — 1 and e + r — 1 m x, 
and L, L ... (L) are of r dimensions in the (a) set, and of (e 4- r) dimensions 
in the (b) set of coefficients, and consequently Lx''^~^ + L'x'^~''~^ + . . . + (L) 
must satisfy the conditions necessary and sufficient to prove its being (to a 
numerical factor pres) a simplified residue to {U, V). 

Thus suppose 

U = QfiX* + a^^a? + a.^x" + a^x + a^,, 

.V = haX" + hiX +60. 

Then, corresponding to the system of which equation (13) is the type, 
we have 

V =h(iX- + \x + h.2, 

xV= boaf + hiof + b^x. 

Again, to form the system of which equation (12) is the type, we write 

boU — (a„a;^ + ais; + Oa) F = 60 («3a; + ^4) — ("o^" + (h^ + a^) (biX + b^) 

= — a^b-i^a? — (a^bz + Oih) «^ + (ho-i — a-ib^ — aj)i) x + (6„a4 — anh), 

(bf,x + &i) U— {a^x^ + «!«- + a.x + a^ V= {b^x + b^) cu — («„«' + «!»■ + a^a; + a^) b^ 

= — ttobiOe' — aibnof + {b^ai — a^h) x + (biQi — Ka^. 



57] 



of tivo Algebraical Functions. 



447 



Combining the two equations of the first system with the first of the second 
system, we obtain the first simplified residue Lx-\-L' , where 



L = 



0, 6o, &i 



and 



n = 



0, \, h 

L' = &„, b„ 

a„bi, a^b^ + aibi, a^b^ — &o o-i 
By again combining the two equations of the first system with both of the 
second system, we liave the determinant 

0, bo, b„ 6, 

h, b„ h, 

tto^i, a^bi + aibi, aib2 + a«hi — bia3, a^b^—bQa.i 
a„&2, a^bo, a.^bn — b^ai, ao62 — tt^&i 

which is the last simplified residue, or in other terms, the resultant to the 

system U, V. 

Art. 7. It is most important to observe that the Bezoutian matrix to two 
functions of the same degree (n) is a symmetrical matrix, the terms similarly 
disposed in respect to one of the diagonals being equal. 

Thus retaining the notation of Art. 5, so that 

(0, 1) = a/3 - ba, (1, 2) = by- c/3, (2, 3) = cS - dy, 
(0, 2) = ay- ca, (1, 3) = 6S - d^, &c. 

(0, 3) = aS-da, &c. 

&c. 
when n = 1 the Bezoutian matrix consists of a single term (0, 1) ; 
when n = 2, it becomes 



when n = 3, it becomes 



(0, 1) 


(0, 2) 




(0, 2) 


(1, 2); 




(0, 1) 


(0,2) 


(0, 3) 


(0, 2) 


/(O, 3)\ 
V(1.^2)/ 


a, 3) 


(0,3) 


(1> 3) 


(2, 3); 



448 On a Theory of the Syzygetic Relations [57 



when n = 4, it becomes 



(0, 1) (0, 2) (0, 3) (0, 4) 
-(0, 3)\ /(O, 4)N 



(0, 2) + + (1, 4) 

,(1, 2)7 \{1, 3), 

^(0, 4)\ /(I, 4)N 

(0, 3) ( + + ) (2, 4) 



,(1, 3)/ \(2, 3), 
(0, 4) (1, 4) (2, 4) (3, 4); 



when n= 5, it becomes 




(2, 5) 



(3, 5) 



(0, 5) 



and so foi-th. Every such square it is apparent may be conceived as a sort 
of sloped pyramid, formed by the successive superposition of square layers, 
which layers possess not merely a simple symmetry about a diagonal (such 
as is proper to a multiplication table), but the higher symmetry (such as 
exists in an addition table), evinced in all the terms in any line of terms 
parallel to the diagonal transverse to the axis of symmetry being alike*. 
Thus for n = 5, the three layers or stages in question will be seen to be, 
the first — 



(0, 1) 


(0, 2) 


(0, 3) 


(0,4) 


(0,5) 


(0, 2) 


(0, 3) 


(0,4) 


(0, 5) 


(1, 5) 


(0, 3) 


(0,4) 


(0, 5) 


(1. 5) 


(2, 5) 


(0, 4) 


(0, 5) 


(1: 5) 


(2, 5) 


(3, 5) 


(0, 5) 


(1, 5) 


(2, 5) 


(3. 5) 


(4, 5); 



* A square arrangement having this kind of symmetry, namely, such as obtains in the 
so-called Pythagorean addition table as distinguished from that which obtains in the multiplica- 
tion table, may be universally called Persymmetric. 



57] of two Algebraical Functions. 449 

the second — 



(1, 2) 


(1> 3) 


(1, 4) 


(1>3) 


(1> 4) 


(2, 4) 


(1. 4) 


(2, 4) 


(3, 4); 



and the third — • 

(2, 3). 

In general, when n is odd, say 2p + 1, the pyramid will end with a single 
term {p, {p+ 1)), and when even, as 2jo, with a square of four terms, 

((p-2), (p-1)), {{p-2),p) 
{(p-2),p), ((p-np). 

Each stage may be considered as consisting of three parts, a diagonal set of 
equal terms transverse to the axis of symmetry, and two triangular wings, 
one to the left, and the other to the right of this diagonal ; the terms in each 
such diagonal for the respective stages will be 

(0, n), (1, n - 1), (2, {n - 2)) ...{p,{p + 1)), 

p being „ — 1 when n is even, and — - — when n is odd. 

If we change the order of the coefficients in each of the two given functions, 
it will be seen that the only effect will be to make th^ left and right triangular 
wings to change places, the diagonals in each stage remaining unaltered. 
The mode of forming these triangles is an operation of the most simple and 
mechanical nature, too obvious to need to be further insisted on here. 

Art. 8. When we are dealing with two functions of unequal degrees, 
n and n + e, we can still form a square matrix with the coefficients of the 
two systems of e and n equations respectively, but this will no longer be 
symmetrical about a diagonal ; it is obvious, however, that if we treat the 
function of the lower degree, as if it were of the same degree as the other 
function, which we may do by filling up the vacant places with terms 
affected with zero coefficients, the symmetry will be recovered ; and it is 
somewhat important (as will appear hereafter) to compare the values of the 
Bezoutian secondaries as obtained, first in their simplest form by treating 
each of the two functions as complete in itself, and secondly, as they come out, 
when that of the functions which is of the lower degree is looked upon as a 
defective form of a function of the same degree as the other. A single 
example will suffice to make the nature of the relation between the two sets 
of results apparent. 

Take 

fx = a os^ + h os^ + CO? -\- dx + e, 

(j)x = x^ + x^ + lyx- + Bx + 6. 
s. 29 



450 On a Theory of the Syzygetic Relations [57 

The general method of Art. 7 then gives for the Bezoutian matrix 
0, Or/, aB, ae 



'■0 

©■ 



be 
aS, { + ) , ( + ) , ce — ey 



ae, be, ce — ey, de — eB. 

We shall not affect the value either of the complete determinant, or 
of any of the minor determinants appertaining to the above matrix, by 
subtracting the first line of terms, each increased in the ratio of 6 : a, from 
the second line of terms respectively ; the matrix so modified becomes 



0, 


ay, 


aB, 


ay, 


aB, 


ae. 


aB, 


ae + bB, 


\cB — dy_ 


ae, 


be, 


Ce — ey, 



ce — ey 

de — eB. 

Again, adopting the method of Art. 6, we should obtain the matrix 

0, 7, B, e 

y, 8, e, 

/ be \ 
B, ae—bB, I + \ , ce — ey 

\cB — dyj 

ae, be, ce — ey, de — eB. 

Hence it is appai-ent that the secondary Bezoutians obtained by the 
sj'mmetrizing method will differ from those obtained by the unsymmetrical 
method by a constant factor a" ; and so in general it may readily be shown 
that the secondary Bezoutians, by the use of the symmetrizing method, will 
each become affected with a constant irrelevant factor a", where a is the 
difference of the degrees of the two functions, and a the leading coefficient 
of the higher one of the two. When a is taken unity, the Bezoutian 
secondaries, as obtained by either method, will of course be identical. 

Art. 9. There is another method* of obtaining the simplified residues 
to any two functions TJ and V of the degrees n and n + e respectively, which, 

* Originally given by myself in the London and Edinburgh Philosophical Magazine, as long 
ago as 1839 or 1840 [p. 54 above] ; and some years subsequently in unconsciousness of that 
fact, reproduced by my friend Mr Cayley, to whom the method is sometimes erroneously 
ascribed, and who arrived at the same equations by an entirely different circle of reasoning. 



57] of two Algebraical Functions. 451 

although less elegant, ought not to be passed over in silence. This method 
consists in forming the identical equations (of which for greater brevity the 
right-hand members are suppressed) 

F=&c. 
xV = &c. 

a;e-iF=&c. 

U" = &c. 

afV = &c. 

af+T-V = &c. 

a^+''F=&c. 

&c. = &c. 

a;"-' U = &c. 

a;''+"-iF= &c. 

If we equate the right-hand members of (e -I- 2i.) of the above equations 
. to zero, and then eliminate dialytically the several powers of x from 3;"+"+'"^ 
to a;"~'+' (both inclusive), the result of this process will evidently be of (e -|- 1) 
dimensions in respect of the coefficients in V, of ( dimensions in respect 
of the coefficients in U and of the degree «""' in a;; it will also be of the 
form 

{A+Bx + ...+Laf-') U + {F+Gx + ... + Qx'+'-^) V, 

and by virtue of Art. 2, must consequently be the ith simplified residue to 
the system U, V. 

Art. 10. The most general view of the subject of expansion by the 
method of continued division, consists in treating the process as having 
reference solely to the two systems of coefficients in U and V, which them- 
selves are to be regarded in the light of generating functions. To carry out 
this conception, we ought to write 

U= ao + a^y + a^y"- -\- a-^xf -\- &c. ad inf. 

V=bo + biy + b^y^ + b-^y^ + &c. ad in/., 
and might then suppose the process of successive division applied to JJ and 
V, so as to obtain the successive equations 

U -M,V +R, = 0, 

V - M,R, + R,^0, 

R,-MA + R, = 0, 

&c. &c., 

29—2 



452 On a Theory of the Syzygetic Relations [57 

il/i, ifa, M^, &c. being each severally of any degree whatever in y, and in 
general the degree of y in M^ being any given arbitrary function ^ (t) of t. 
The values of the coefficients of the residues R^, B^, R-i ... , or o{ these forms 
simplified by the rejection of detachable factors, become then the distinct 
object of the inquiry, and will, of course, depend only upon the coefficients 
in U and V and the nature of the arbitrary continuous or discontinuous 
function (f) (t), which regulates the number of steps through which each 
successive process of division is to be pursued. Following out this idea in a 
particular case, if we again reduce our two initial functions to the forms 
previously employed, and write 

f/" = ao*" + «!«;""' + &c. 

V = \x"- + 6ia;"-i + &c. ; 

and if, instead of making, according to the more usual course of proceeding, 
the divisions proceed first through one step and ever after through two steps 
at a time, which is tantamount to making ^1 = 1, (1 + o)) = 2, we push each 
division through one step only at a time, and no more (so that in fact <p (t) 
is always 1), we shall have 

U - m, F + iii = 0, 

i?! — wis jRa + -Rs = 0. 

i?2 — iriixRs + Ri=0, 
&c. &c., 

mi, ma, m^, &c. being functions of the coefficients only of U and F; and it is 
not without interest to observe (which is capable of an easy demonstration) 
that the simplified residues contained in R^, i?,, &c., found according to this 
mode of development, will be the successive dialytic resultants obtained 
by eliminating the (t — l)th highest powers of x between the i first of the 
system of annexed equations (supposed to be expressed in terms of x) 

cr=o, 

F = 0, 

xU=0, 

xV=0, 

a?U=0, 

a;2F=0, 

&c. = &c 
x»-^U= 0, 
a;«-iF= 0. 



57] of tivo Algebraical Functions. 453 

If we combine together 2i + 1 of the above equations, the highest power of 
X entering on the left-hand side will be a;""^', and we shall be able to eliminate 
2i of these factors, leaving «""' the highest power remaining unelimiuated. 
If we take 2i, that is i pairs of the equations, the highest power of x appear- 
ing in any of them will be a;"+*~S and we shall be able to eliminate between 
them so as still to leave ajn+i-i-i^i'-D^ that is «"""* as before, the highest power 
of X remaining uneliminated ; and it will be readily seen that such of the 
simplified residues corresponding to this mode of development as occupy the 
odd places in the series of such residues, will be identical with the successive 

simplified residues resulting from the ordinary mode of developing ^ under 
the form of a continued fraction. 

Art. 11. It has been shown that the simplified residues oi fx and <px 
resulting from the process of continued division are identical in point of 
form with the secondary Bezoutians of these functions, but it remains to 
assign the numerical relations between any such residue and the corre- 
sponding secondary. 

To determine this numerical relation, it will of course be sufficient to 
compare the magnitude of the coefficient of any one power of x in the one, 
with that of the same power in the other ; and for this purpose I shall make 
choice of the leading coefficients in each. In what follows, and throughout 
this paper, it will always be understood that in calculating the determinant 
corresponding to any square the product of the terms situated in the diagonal 
descending from left to right will always be taken with the positive sign, 
which convention will serve to determine the sign of all the other products 
entering into such determinant. Now adopting the umbral notation for 
determinants*, we have, by virtue of a much more general theorem for 
compound determinants, the following identical equation : — 

/(ha^as ...a,n-i\ fa-^aM^ ... «„,+!> 
\ «! a2 Os ■•• am-J VaiHaas ... a,„+J 

_ /Uia^as ... a,„_ia,„\ /aia,, ... am-ia,n+i\ 

Ka^aMs ... am-lO-m) K'^l'^ •■■ am-l«m+l/ 
tti tta as • • • 0,111-1 C'm \ /«! iXo . . . a„i_i «m+i\ 

Kia^Os ... a„,_iam+i/ VMiHo ... am-ia»)i / ' 



and consequently 



fa^a^tts . . . flm-A fClia^a.s . . . Clm-iO'mnm+i 

\^ a^a^a-i . . . a,„_J ■ VoiMaas • ■ • aiH-iam«™+i 



aitt^a^ . . . a,„_ia„A fa-, a.,... a,„_ia„i+i\ /«!«. 



i,n y 



See Lotulon and Edinburgh Philosophical Magazine, April 1851 [p. 242 above]. 



454 



071 a Theory of the Syzygetic Belations 



[57 



and consequently when 






U,a„ ... «,„_,, 



and 



- =0, 



will have different algebraical signs, it being of course understood that all the 
quantities entering into the determinants thus umhrally represented above 
are supposed to be real quantities. This theorem, translated into the ordinary 
language of determinants, may be stated as follows : — Begin with any square 
of terms whether symmetrical or otherwise, say of r lines and r columns: let this 
square be bordered laterally and longitudinally by the same number r of new 
quantities symmetrically disposed in respect to one of the diagonals, the term 
common to the superadded line and column being filled up with any quantity 
whatever ; we thus obtain a square of (r + 1) lines and columns ; let this 
be again bordered laterally and longitudinally by (r+1) quantities symme- 
trically disposed above the same diagonal as that last selected, the place 
in which this new line and column meet being also filled up with any arbitrary 
quantity ; and proceeding in this manner, let the determinants corresponding 
to the square matrices thus formed be called D,-, i),+i> Dr+2---'- this 
series of quantities will possess the property, that no term in it can vanish 
without the terms on either side of that so vanishing having contrary signs. 
Thus if we begin with a square consisting of one single term, we may suppose 
that by accretions formed after the above rule it has been developed into 
the square (M) below written, and which of course may be indefinitely 
extended : — 

a, I, 



p, s, 
I, b, n, q, t, 
in, n, c, r, u, 
p, q, r, d, V, 
s, t, u, V, e. 
Here D^, D^, D^, D^, A, -^s will represent the progression 



(M) 



a, I 
I, b 


' 


a, I, ni 
I, b, n 
m, n, c 



a, I, VI, p 




I, h, 11, q 




m, n, c, r 


' 


p, q, r, d 





a, 


I, 


m. 


P' 


s 


I. 


b, 


n. 


9' 


t 


m, 


«; 


c, 


r, 


u 


P> 


9. 


f, 


d. 


V 


s, 


t, 


u, 


«, 


e 



(H) 



57] 



of two Algebraical Functions. 



455 



so if we use the matrix 



a, 


I, 


m, 


p, s 


, 








I', 


h, 


n, 


q, t 










m, 


n, 


c, 


r, u, 








P' 


?> 


J', 


d, % 


. 








s, 


t, 


u, 


V, e 


, 








A 


represeni 


ing 


















a, 


I, 


m, 


P 


a, 


I, 


m 




I', 


b, 


n, 


q 


I', 


h, 


11 


^ 


















m, 


11, 


c, 


r 


m, 


n, 


c 




















P> 


(h 


r. 


d 



will possess the property in question ; the line and column I, b; V, b not 
being identical, the first determinant -Do representing unity must not be 
included in the progression. 

We shall have occasion to use this theorem as applicable to the case of a 
matrix symmetrical throughout, and we may term the progression (11), above 
written, a progression of the successive principal determinants about the 
axis of symmetry of the square matrix (M), and so in general. Now it is 
obvious that the leading coefficients of the successive Bezoutian secondaries 
are the successive principal determinants about the axis of symmetry of the 
Bezoutian squares; they will therefore have the property which has been 
demonstrated of such progressions ; to wit, if the first of them vanishes, the 
second will have a sign contrary to that of + 1 ; if the second vanishes, 
the third will have a sign contrary to that of the first, and so on. 

Art. 12. Now let fx and <l>x be any two algebraical functions of x with 
the leading coefficients in each, for greater simplicity, supposed positive : 

and in the course of developing %- under the form of an improper continued 

/^ 
fraction by the common process of successive division, let any two consecutive 
residues (the word residue being used in the same conventional sense as 
employed throughout) be 

Ax' + 5a;'-> + (?«'-= + &c. 

5'a;'-i + C V-' + D'a;'-' + &c. 

The residue next following, obtained by actually performing the division and 
duly changing the sign of the remainder, will be 

(AC ^\ G'l 



( AD' 
\B' ■ 



C 



-{^-^ 



' + &c.. 



456 On a Theory of the Syzygetic Relations [57 

which is of the form 

Thus the leading coefficients in the complete unreduced residues will be 

A, B', ^^{B'M-AG'"-], 

and when reduced by the expulsion of the allotrious factor will become 
A, B', B'M — AG'-, and consequently, when B' the leading coefficient of one 
of the simplified residues vanishes, the leading coefficients of the residues 
immediately preceding and following that one will have contrary signs. 

First, let fx and <^x be of the same degree. As regards the numerical 
ratio of each Bezoutian secondary to the cori-esponding simplified residue, 
it has been already observed that there are always unit coefficients in the 
latter of these, and the same is obviously true of the former ; hence if we 
call the progression of the leading coefficients of the simplified residues 

R^, R^, Rs, R„ &c., 

and that of the leading coefficients of the Bezoutian secondaries 

B„ B^, B„ B„ &c., 
we have 

B, = ±R„ Bo=±R2, B, = ±R„ B, = ±Ri, &c. 

It may be proved by actual trial that B^ — R^ and B., = R.,. Moreover, 
since the signs are invariable, and do not depend upon the values of the 
coefficients, we may suppose ^2 = (which may always be satisfied by real 
values of the quantities of which B^ is a function) ; we shall also, therefore, 
have iJ, = 0, and consequently B^ has the opposite sign to that of B^, and R, 
the opposite sign to that of R^, which is equal to B^: hence when B^ = Q, 
Bs and R, are equal, and consequently are always equal ; in like manner we 
can prove that R^ and ^4 have the same sign when R^ and B3 vanish, and 
consequently are always equal, and so on ad libitum, which proves that the 
series B^, B^, ... Bn is identical with the series R^, R^, ... Rn, and con- 
sequently that the Bezoutian secondaries are identical in form, magnitude 
and algebraical sign with the simplified residues. 

Secondly, when fx and <^x are not of the same degree, it has been 
shown that the secondaries formed from the non-symmeti'ical matrix corre- 
sponding to this case will be the same as those formed from the symmetrical 
matrix corresponding to fx and <I>a; (where (i>x is ^x treated by aid of 
evanescent terms as of the same degree as fx), with the exception merely 
of a constant multiplier (a power of the leading coefficient of fx) being 
introduced into each secondary. By aid of this observation, the proposition 



57] of two Algebraical Functions. 457 

established for the case of two functions of the same degi-ee may be 
readily seen to be capable of being extended, from the case of / and <p 
being of equal dimensions in x, to the general case of their dimensions being 
any whatever. 

Art. 13. Before closing this section, it may be well to call attention to 
the nature of the relation which connects the successive residues of fa and 
ipx with these functions themselves, and with the improper continued 

fractional form into which ^ is supposed to be developed in the process of 
obtaining these residues. 

If (jix be of n degrees, and fa of n + e degrees in x, we shall have 
^_ J^ ]_ J_ l^ 

fa~ Qi- q-i-q^-'" qn' 

where Qj may be supposed to be a function of x of the degree e, and 
^2, qz, ••• qn, are all linear functions of x; the total number of the quotients 
Qi, q„, ... qn being of course n when the process of continued division is 
supposed to be carried out until the last residue is zero. Upon this supposi- 
tion the last but one residue is a constant, the preceding one a function of x 
of the first degree, the one preceding that a function of x of the second 
degree, and so on. 

Let us call the residue of the degree t in x, Sr^ ; it will readily be seen 
that the successive complete residues arranged in an ascending order will be 

^0, Xqn, ^0 (qn-iqn - 1), S^o (qn~2qn-iqn - qn-^ - qn), &C., 

being in the ratios 

1, Qn, Qn-i , qn-" , &C. 

qn qn-i - qn 

Again, we shall have in general 

AJ-L,^ = X, (15) 

A, being an integral function of x of the degTee n— u — l, and L^ an integral 
function of x of the degree (n + e) — t — 1 ; and it is easy to see that the 
successive convergents to the continued fraction 

111;! 

Qi-q-.-q^- 

have their respective numerators and denominators identical with those of 
the fractions 

A^, A^ A^3 

J^n—i J-'n-2 J^n-3 



458 On a Theory of the Syzygetic Relations [57 

Adopting the language which I have frequently employed elsewhere, 
I call ^j a syzygetic function, or more briefly a conjunctive of / and (f>, and 
A, and L, may be termed the syzygetic factors to ^^ so considered. If we 
divide each term of the equation (15) by the allotrious factor {M), we have 

where R, is the tth simplified residue to (/, <f>); and if we call -ri^Tt, and 

■p-= <i, so as to obtain the equation 

Tj-t,4> = R„ (16) 

we see that -', the fraction formed by the component factors to any simplified 

residue of (/, </>), will be identical in value (although no longer in its separate 

terms) with one of the corresponding convergents to ^, exhibited under the 

form of an improper continued fraction. I shall in the next section show 
how, not only the successive simplified residues, but also the component 
syzygetic factors of each of them, and consequently the successive con- 
vergents, may be expressed in terms of the roots of the two given functions. 

Since the preceding section was composed the valuable memoir of the 
lamented Jacobi, entitled "De Eliminatione Variabilis e duabus Equationibus 
Algebraicis," Crelle, Vol. xvi., has fallen under my notice. That memoir is 
restricted to the consideration of two equations of the same degree, and the 
principal results in this section as regards the Bezoutic square and the 
allotrious factors applicable to that case will be found contained therein. 
The mode of treatment however is sufficiently dissimilar to justify this 
section being preserved unaltered under its original form. 



Section II. 

On the general solution in terms of the roots, of any two given algebraical 
functions of x, of the syzygetic equation, which connects them with a third 
function, whose degree in x is given, hut whose form is to he determined. 

Art. 14. Let / and <^ be two given functions in x of the degrees m and 
n respectively in x, and for the sake of greater simplicity let the coefficients 
of the highest power of a; in y and <^ be each taken unity, and let it be 
proposed to solve the syzygetic equation 

tJ- t,^ + ^, = 0, (17) 



57] of two Algebraical Functions. 459 

where ^^ is given only in the number of its dimensions in x, which I suppose 
to be t; but the forms of t„ t^, S-^ are all to be determined in terms of 
/i], /(„ ... hm the roots of/" and t/j, t]^ ... rjn the roots of (p. 

I shall begin with finding ^^ ; and before giving a more general represen- 
tation of ^i, I propose now to demonstrate that we may make 

^, = 2 {P,„ 5, ... ,, X {x - h,) {x - A J ...{x- k,)}, (18) 

where Pq„q.,...r/, is used to denote 

' (K+i - ''O ('''r.+i - V2) ■ ■ ■ {hg^+, - Vn)\ 

^ (K-i2 - Vi) {K+2 -V2)... {\+^ - Vn) 



^ X (Kn - Vi) {l>qn, -V2) {hq^ - Vn) I 

B (jkg^, hq^ ... hq) denoting any rational symmetrical function whatever of 
the quantities preceded by the symbol R, and q^, q«...qn q,+i ...qm being any 
permutation of the m indices 1, 2 ... m. 

Sui3pose/= and ^ = 0, then x is equal to one of the series of roots 

/l], h.,... hm, 

and also to one of the series of roots 

Suppose then that 

X = ha = Viii 

and consider any term of^,. 

If in any such term a is found in the series 51, q^ ... q,,, then 

(X - hq) (X - hq^ ...(«- hq) = 0. 

But if not, then x must be found in the complementary series 

and consequently Pq„q,...q, will contain a factor K-v^ and -Pa,,?^... 3^ = 0; in 
every case therefore 

PqnQ. ... 7. X (^ - K) ('^ - K) •••(«- K) = 0- 

Therefore ^, as expressed in equation (18) is a syzygetic function of / 
and (f> ; and we have found a function of the tth degree in *•, and of course 
expressible by calculating the symmetric functions as a function only of x and 
of the coefficients of/ and ^, which will satisfy the equation 



460 On a Theory of the Syzygetic Relations [57 

It will be remembered that by virtue of Art. 2 we know a jjriori that all 
the values of ^^ satisfying this equation are identical, save as to an allotrious 
factor, which is a function only of the coefiScients in / and (f>. 

It is clear that we may interchange the h and rj, m and n, and thus 
another representation of a value of S-i satisfying the equation (17) will be 

S. = 2^(77,,, Vq,---V%) X 

(Vq,+. - hi) iv<j,+„ - K) . . . {Vq,+„ - hm) 



; iVq,, - !h) iVq^ - ho) ■ ■ ■ iVqn " K) ^ 

Art. 15. If we employ in general the condensed notation 
p, VI, 71 ... pi 

\_\, fX, ... V J ' 

to denote the product of the differences resulting from the subtraction of 
each of the quantities \, fi...v in the lower line from all of those in the 
upper line I, m, n...p, the two values above given for S-^ may be written 
under the respective forms 



'ZRih,^,h,.,...hg). 



and 2-B (»?f, , Vi2--- vO • 



-K^. 


K+ 


-Vi< 


V2 ■ 


^\.. 


%+ 


Ih, 


h,. 



(X - hq^ 



)(x-hg,) ... {x-hq), 



{X - r)^^) (X - 7]^^) ...{X- 7]^), 



in each of which equations disjunctively and in some order of relation each 
with each 

qi, q., qs... g',„ = l, 2, 3...m, 

and ^1, |„^3.--^»=1, 2, 3...«. 

These two forms are only the two extremities of a scale of forms all equally 
well adapted to express S^^ ; for let v and v be any two integers so taken as 
to satisfy the equation 

and let R ( ; ), where the dots denote any quantities whatever, be 

used to denote a rational function which remains unaltered in value when any 
two of the quantities under either of the two bars are mutually interchanged, 
then we may write 



% = t 



R(hq^, hq, 



ha 



' ^f.'^'s.--- V 



-''l.«' \ 



x(x- hg) (x - hgj ...{x- hg^) x{x- r]^^) (x - v^, 



-7/,)...(«-1?,)) 



(19) 



57] 



of two Algebraical Functions. 



461 



For if, as above, we suppose a; = Aa = t?^, any term of S^^ in which q^, q.,... q„ 
comprise among them a, or in which ^], ^3... ^„ comprise among them w, 
will vanish by virtue of the factors 

(w - lij (x - kj . . . (.r - hg^) x(x- 97fj) (x - 7,Q ...{x- 7]^ J ; 
but if neither a nor eo is so comprised, then a must be one of the terms 
in the complementary series g'^+i. ^d+s, ••• qm, and eo one of the terms in the 
complementary series ^,/+i, ^„+2 ... f„, and therefore one of the quantities 
^*?B+i' ^<!v+2 ••• ^Vm '^^^^ equal one of the quantities rj^ ,rj^ , ••■ "^f > ^^^ '^°^' 
sequently the term of '^^ in question will vanish by virtue of the factor 

""*""• "' vanishing. In either case therefore every term included 

within the sign of summation vanishes when x=h^ = 7j„, that is, whenever 
fx — and (j)x=0. Hence S-„ as given by equation (19), will satisfy 
the syzygetic equation t,/— t,(j) + ^, = for all values of v and v which 
make v + v = l, and for all symmetrical forms of the function denoted by the 
symbol R ( ; ). 

Art. 16. I shall now proceed to show how to assign the arbitrary 
function whose form is denoted by this symbol in such a manner as to make 
^, become identical with a simplified residue to / and (f). To this end I take 



fori? (/(,,, hq^...hq^; Vh' Vh 



R = 



rjij the value 
h„. ... h 






'??. 






we shall then have 



Qv+i' 1v+2 ' 






(20) 



^. = 2- 






'^h'^^h-^'i. 



J L 






"^U 



] 



U*„+, > K+. ■ ■ ■ Kn^ L'/i,+, , Vl^^^ ...v^J 

X {(«-''«,)(a'-^?,)---(«-MK«'-'?f.)(«-%)---(*-'?U}- (21) 
I shall first show this sum of fractions is in substance an integral function 
of the quantities h^, lio_...lhn\ Vu Vi---Vn- For greater conciseness write 
in general x — h = E,x — r) = H; we have then, since 

~H(^,Hi^... Hi^~\ T-^f -4-1 ' -^f .+2 • • • ^^n 



X^t 



[H,,, 
K. 



Ea. ...En 



r^w>ifw2-^^.1 



E„ 



-^2«+2"-^'?: 



Eo. 



Ea. 






.E„. 



E„ Hi, ... H, 



"Iv 



(22) 



462 



On a Theory of the Syzygetic Relations 



[57 



On reducing the fractions contained within the sign of summation to a 

N 
common denominator, ^, will take the form ^^—^< where D will be the 

product of the ^m(m-l) differences oi E^, E„...Em subtracted each from 
each, and A the corresponding product of the differences inter se of 
Hi, H^...Hn- Hence, unless the sum in question is an integral function 
of the E's and H's it will become infinite when any two of the E series, or 
any two of the H series of quantities are made equal. Suppose now E-, = E^; 
the terms in (22) which contain E^ - E^ in the denominator will evidently 
group themselves into pairs of the respective forms. 



{E,E,^...EJx(H(,Hi,...Hi^)x 



'El, Eq^ ■■■Eg^ 



1 [E 



^1v+-2---^"- 



!)+2 ■ • • ^1m~\ 



Ei, Eq.^ 



[_E._, 



En, 



-iff,, ih, ...H,;\ 



and 



{E.E,,...E,)x{E,,E,,...HOy~\„ £ jr 



E._, E„. 



E, 



E E 



...E„. 






Ie„e,^^^...e,J L^,.,^„ 



"fff., Hi. -a,^ 






the sum of this pair of terms will be of the form 



E, 



E, 

Rp,, H(„...E, 



Kk.. 



Hi 



H 



E,-E„ 



E, 

Hq^+l'Hq^+2---Hq, 



inJ [_ 



p 


[E, 1 [E, 


••^J 


Q 


E^ — El En 





where Q, it may be observed, does not contain H^ — H^, so that ^ remains 

finite when Hi = H2. 

The above pair of terms together make up a sum of the form 

P 1 ^ (El, E,) ■^E^-(f> {E„ E\) ■r^Ei 
QEi-E^ -fEiX-fEn 

which, as the numerator of the third factor vanishes when E^ = E^, remains 
finite on that supposition. Hence the whole sum of terms in (22) which is 



57] of two Algebraical Functions. 463 

made up of such pairs of terms, and of other terms in which E^ — JE., does 
not enter, remains finite when E^ — E^= 0, and therefore generally when 
D = 0, and similarly when H^ — H„ = 0, and therefore also when A = ; 
hence the expression for ^^ in (22) is an integral function of the E and H 
series of quantities, as was to be proved. 

Art. 17. Let us now proceed to determine the dimensions of the coeffi- 
cient of «', the highest power of x in this value of ^., when supposed to be 
expressed under the form of an integral function (as it has been proved to be 
capable of being expressed) of hi, Aj . . . A^ ; Vi> Vi ■•• Vn', oo. 

This coefficient is the sum of fractions the numerators of each of which 
consist of two factors, which are respectively of « x z/ and of (m — v)x (n — v) 
dimensions in respect of the two sets of roots taken conjointly, and the 
denominators of two factors respectively of v {in — v) and v (n — v) dimen- 
sions in respect of the same. 

Consequently, the exponent of the total dimensions of the coefficient in 
question 

= vv + (m —v){n — v) — v (m — v) — v{n — v) 
= {m — v — v)y. {n — v — v) 
= (m — t) (w — t), 

and thus is seen to depend only on the degree t in a; of S^„ and not upon the 
mode of partitioning i into two parts v and v, for the purpose of representing 
^„ by means of formula (19). 

Art. 18. I shall now demonstrate that, every form in this scale (to a 
numerical factor pres) is identical with a simplified residue to /, <^, of the 
same degree i in x. Any such simplified residue is, like &„ a syzygetic 
function, or to use a briefer form of speech a conjunctive of/, ^; and if we 
agree to understand by the " weight " of any function of the coefficients of 
/ and ^ its joint dimensions in respect of the roots of / and ^ combined, 
I shall prove, — first, that any simplified residue of/ and ^ of a given degree 
in X is that conjunctive, whose weight in respect of the roots of /and <^ 
is less than the weight of any other such conjunctive ; and second, that '^„ 
as determined above (in equation 22), is of the same weight as the simplified 
residue, and can therefore only differ from it by some numerical factor. 
For the purpose of comparison of weights, it will of course be sufficient 
to confine our attention to the coefficients of the highest power in *■ (or 
any other, the same for each) of the forms whose weights are to be compared. 

Suppose / to be of m dimensions, and ^ to be of n dimensions in x ; 
and let m = n + e. 



464 On a Theory of the Syzygetic Relations [57 

Suppose Af+L^ = Ax' + Bx'-^ + ...+K, (23) 

A = X^xi + X^xi-^ + . . . + X^, 

the number of homogeneous equations to be satisfied by the ^ 4- 1 quantities 
X„, Xi...X,, and the g + e + 1 quantities ?„, li---lq+e will be m + g— i, and 
therefore g' + 1 and q + e+1 taken together must be not less than «i + g — i + 1, 
that is 2^+6 + 2 must be not less than g + m — t + 1, that is q not less than 
m — i — e — \ ; and if this inequality be satisfied 2g + e + 2 — (g + m — t + 1) + 1, 
that is q + i + e — m+2 will be the number of arbitrary constants entering 
into the solution of equation (23). 

If q be greater than (» — 1), let q = {n — l) + t; and let 
(A) = X„a;''-i + Xia;"-2 + . . . + X„_, , 
(L) = kx^+''-^ + ?ia;«+^-= + . . . + le+n-, ; 
and let (A), (Z) be so taken as to satisfy the equation 

{h)f+{L)4>=Ax'-^-Bx'~'' + ...+K; 
and make 5 = (A) ■'r {f + gx + ... + Aa;*~^) (f>, 

X={L)-if+gx + ...+hx^-^)f, 
f, g ... h being arbitrary constants ; then 

E/+ Z(/) = (A)/+ (X) (]i = Ax' + £«'-' + ...+K. 
Now the total number of arbitrary constants in the system (A) and (L) 
will be n — 1 + I + e — m + 2, that is t + 1 ; hence the total number of arbitrary 
constants in H and X will he o + 1 +t, that is q — n + 1 + 2, which is equal to 
q + i + e—m + 2, the number of arbitrary constants in the most general values 
of A and L. Hence {A = H, L = X] is the general solution of the equation 
Af+L<j)=Ax'- + £x'~^ + ...+K; and consequently the most general form 
oi Ax' + Bx'~^ + ...+ K, which is evidently independent of the (t) arbitraiy 
quantities /, g ... h, will contain the same number of arbitrary constants 
as enter into the system (A) and (L), that is t + 1. 

Art. 19. Let us now begin with the case of greater simplicity when 
m = n, that is e = ; and let us revert to the sj'stem of equations marked (10) 
in Section I., in which U and V are to be replaced by /and cj). 

First, let t = ?i— 1, then t + 1, the number of arbitrary quantities in the 
conjunctive, is n. 

From the system of equations (10) we have, for all values of pi, p.^, ps... pn, 

(PiQo + PiQi + ■ ■ . + pnQn-i)f 
- (piPo + p2-Pi + • • • + /3«-Pn-i) </> 
= {p^K^ + p,,K, + ...+pn n-iK,) X^-' + fee, 



57] of tioo Algebfaical Functions. 465 

and consequently the most general value of ^„_] in the equation 

T«_i/- tn-i4> + ^,1-1 = 0, 

where ^„_i = Ax^'-~^ + Bx^-- + ... ■\- L, 

will be obtained by making 

Tn-i = jOiQo + JO2Q1 + . • ■ + PnQn-i, 
tn-i = — PiPo — P«Pi ■■■ — PnPn-i, 

which solution contains n, that is the proper number of arbitrary constants. 

Again, if i=n — 2, i + l =n — l, which will therefore be the number 
of arbitrary constants in the most general value of ^„_2 in the equation 

Tn-if — tn-ii) + ^«-2 = 0. 

This most general value of ^„_2 is therefore found by making 

T„_2 = /s'lQo + p\Q-^ + . . . + p'nQn-i, 
tn—i ^ P iJ- P2 -^1 • • • P n "n—\ > 

where p\, p\...p'n are no longer entirely independent, but subject to the 
equation 

p\ K^ + p'„ iZi + . . . + p'„ n-^K^ = 0, 
so as to leave (ri — 1) constants arbitrary. 

We thus obtain ^„_2 = (p'lifs + p'i 1-^2 + ■■■+ p'n ii-1-^2) «"~^ + &c. In like 
manner, and for the same reasons, the most general values of %i-z in the 
equation 

will be found by making 

T„_3 = pi\ Qo + p". Qi+ ... + p"n Q,i-i, 
tn-3 = — P"l Pts- p'lPl---— P"n Pn-l, 

where p"i, p'\ ... p"n are subject to satisfying the two equations 

p'\ Ki + p". ^Ki+ ... + p"n n-iJ^i = 0, 
p'\ K._ + p"n liTa + . . . + p"n n-iK^ = 0, 

SO as to leave {n — 2) constants arbitrary ; and we thus obtain 

^„_3 = {p'\ Ks + p'\ iKs+ ...+ p"n n-J^i) «"~^ + &c., 
and so on, the number of independent arbitrary constants in ^ decreasing 
(as it ought) each time by one unit as the degree of ^ descends, until finally, 
if To/— io^ +^0 = 0, ^0 being a constant, the general value for ^o is found by 
making 

To = (pO Qo + (/32) Qi + . . . + (p«) Q,.-i, 
i„ = - (pO P„ - (p,) P, - . . . - (p„) P„_, , 

s. 30 



466 Oil a Theory of the Syzygetic Relations [57 

where (p,), {p„) ... (p„) are subject to satisfy the (n- 1) equations 
(pO K, + &c. = 0, 
(pO K, + &c. = 0, 



(/3i)AVi + &c.=0, 

which gives \ = Kn{p)i + i^n (p)o + . . . + n-iKn {p)n . 

Now evidently the lowest weight in respect to the roots of U and V that 
can be given to (pi-STj + p^ \Ki + . . . + p„ n-iJ^i) «'*~^ + &c., when the multipliers 
Pi, Ps ••■ Pn are absolutely independent, is found by taking 

Pi = l, P2 = 0, p3 = 0...p„ = 0, 

which makes the weight of the leading coefficient in &„_i, the same as that 
of ^1, that is 1. 

Again, when one equation, 

p'l K, + p'o,K,+ ...+ p'n n-iKi = 0, 

exists between the (p)'s, the lowest weight will be found by making 

p'i=i-^i> p'i = -Ki, p'3 = 0, p'4= ...p'„ = 0, 
which makes the weight of the leading coefficient in &„_„ depend on 

which is of the weight 1 + 3, that is 4, in respect of the roots of/ and <^. 

Similarly, S-„_3 will have its lowest weight when its leading coefficient 
is the determinant 

,K„ ,K,, ,K, 

the weight of which is 1+3 + 5 = 9; and finally, the lowest weighted value 
of ^0 is the determinant represented by the complete Bezoutian square ; the 
weight in general of S-„_t being 1+3+ ...+(2i — 1), that is P, or which 
is the same thing otherwise expressed, the weight of the leading coefficient 
of the lowest-weighted conjunctive of / and <^ of the degree t in « is 
{n — (,)(m— t)*. It will of course have been seen in the foregoing demon- 
stration, that the weight of ^Kg [which means S (arbg — a^br), ar, as being the 
coefficients of a;""*", a;™~^ in /, and hy, hg of the same in </>] has been correctly 
taken to be r+ s in respect of the roots of /and ^ conjoined. 

* n and m are supposed equal and i. = n-i. 



57] 



of two Algebraical Functions. 



467 



Art. 20. If now we proceed in like manner with the general case of 
m = n + e, it may be shown, in precisely the same way as in the preceding 
article, that the most general value of any conjunctive of/ and ^ will be a 
linear function of e functions. 




a;" + ai*"-^ + a, «"■-= + . . . + a„ , 
a;"+^ + tti*" + a.x''-^ + . . . + a„a;, 
x'^+^ + aix''^+^ + a.x"' + . . . + UnX", 



and of the n functions, 



&c. 



-=+&c. 
,K,x' 



+ anX^-\ 

■"'+...+ rKn, 
&C. 



and that consequently, if the degree of such conjunctive in x be {n — i), 
it will be of the lowest weight when it is a linear function of the entire 
e upper set of functions, and i of the lower set; and consequently, the 
coefficient of the highest power of x in such conjunctive will be the 
determinant 

K^, K, Ki Ki+e 

,K,, ,K, ,Ki ,Ki+, 

,K„ ,K, ,Ki ,Ki+e 







the weight of which is evidently that of 

K^ X lifs X ^-STs . . . X i_ JiTf X (a^)^ 
that is l + 3 + 5 + ...+(2i-l) + et, 

that is i" + ei, or i (e + i), which is {n — i){m — i) if i = n — i. 



30—2 



468 On a Theory of the Syzygetic Relations [57 

Hence the weight of the leading coefficient in the lowest-weighted 
conjunctive of/ and of the degree t in « is (m — i){n — i), m being the 
degree of/ and n of <f). 

From this we infer that any conjunctive of/ and (p of tlie degree t, 
of which the leading coefficient is of the weight (ni — i)(n — i), all the 
coefficients being of course understood to be integral functions of the roots 
of / and <f>, must, to a numerical factor p7-es, be equivalent to any other 
of the same weight ; and furthermore, any supposed function of x of the tth 
degree which possesses the property characteristic of a conjunctive of vanish- 
ing when / and (}> vanish simultaneously, but of which the weight of the 
leading coefficient would be less than (m — t) (n — t,), must be a mere nugatory 
form and have all its terms identically zero *. 

Art. 21. We have previously shown. Art. 16, that ^, as defined by 
equation (21), is an integral function of the roots / and ^, and vanishes 
when / and </> vanish. Moreover, its weight in the roots has been proved 
to be (m— t) (w - i), and consequently, if by way of distinguishing the several 
forms of ^1 we name that one where i in the equation above cited is supposed 
to be divided into two parts, v and v, &„,„, we have for all values of v and v, 
such that V -t- 7/ is not greater than n, ^^, ^ to a constant numerical factor pris 
identical with the {v + v)th. simplified residue to (/ ^), so that the form of 
^5^^ depends only upon the value oiv-\-v. 

Art. 22. It must be well borne in mind that this permanency of the 
value of ^^,i_„ for different values of v has only been established for the case 
where i can be the degree of a residue to / and (/>, that is to say, when t 
is less than the lesser of the two indices m and n. When t does not satisfy 
this condition of inequality, the theorem ceases to be true. It is clear that 
when m = n and v-\-v = 1171 = 11, ^s,„, which always remains a conjunctive of / 
and ^, can only be a numerical linear function of / and ^ ; and I have 
ascertained when m = 7i on giving to v and v the respective values succes- 
sively (0, n), (1, n - 1), (2, (n - 2)) . . . (n, 0) that 

^„_,,i=/+(»i-i)<^; Vo = '/>- 

Thus, by way of a simple example, let 

/= x^ + ax + h = {x — hi) {x - lu), 

<f) = x- + ax + /3 = {x- k-i) (x — k^), 

* And more generally it admits of being demonstrated by precisely the same course of 
reasoning, that the number of arbitrary parameters in a conjunctive of the degree i, and of the 
weight (m-i)(n-i) + e in t^e roots, cannot (abstraction being supposed to be made of an 
arbitrary numerical multiplier) exceed the number e. 



57] 



of two Algebraical Functions. 



469 



^0,2 = (* — '*l) (* — /i-z) ■ 



%^, = t{x-K){x-k,)^ 



'hji^ 


X t;k 


KK 


kik^ 



■■ (x - hi) (x - Aj) =/, 



that is 



'h; 






1^; 



_ ^ a; - hi j 1 /(« — ki)-{hi — k^) (h^ — k^) \ 
~'^hi- h, \k\ -LA -(x- k^) {hi - k.;) (h, - ki)J 



= S 



X — hi 



{{hi -L)x + [{ki + D L - {h-ih^ + kik,)]} 



/i^ — h^ 

= {x — hi) x + {x— /ig) X — {ki + D X + {hji^ + kik„) 
= [x' - {hi + h^) X + hih} + {x^ - {ki + L) X + kih} 
= («2 + ax + b) + {x^ + ax + ^) 

so we find also ^2,0 = <f>- 



Art. 23. The expression S-„_„, which is universally a conjunctive of/ 
and (j>, continues algebraically interpretable so long as v + v has any value 
intermediate between and m + n; when v + v = we must of course have 
v=0 and j^ = 0, and ^o_„ becomes the resultant of/ and <j> ; when v + v = m + n 
we must also have the unique solution v = m and v = n, and ^,„,k becomes 
necessarily f x ^, which we thus see stands in a sort of antithetical relation 
to the resultant of / and (f), say (/ 0). Nor is it without interest to remark 
that fx(f> = implies that a factor of / or else of (f> is zero ; and (/, (^) = 
implies that if a factor of the one of the functions is zero, so also is a factor 
of the other, that is that a factor of each or of neither is zero. As t increases 
from to n or decreases from m + ?i to m — 1, the number of solutions of the 
equation v + v = i in the one case, and the number of admissible solutions 
of the equation ■?; + t- = i in the other case, which is subject to the condition 
that V must not exceed n, continues to increase by a unit at each step ; 
there being thus n + 1 different forms ^„^y when v + v = n, and the same 
number when v + v = m — 1. For all values of i intermediate between n and 
(m— 1) (both taken exclusively) it is very remarkable that ^^^„ will vanish, 
as I proceed to demonstrate. 



470 On a Theory of the Syzygetic Relations [57 

Art. 24. The weight of the coefficient of the highest power of ^„_„ 
{v + v being equal to t) is (m — i){n — i), and consequently, when t is greater 
than n, and less than m, S-b,„ would contain fractional functions of the roots 
of / and 0, if there were in it a power «', but ^j,, » has been proved to be 
always an integer function of the roots. Hence the coefficient of «' will be 
zero, and so more generally the first power of x in &„,^, of which the coefficient 
is not zero, will be a;'~", subject to the condition (since evidently the weight 
of the several coefficients goes on increasing by units as the degree of the 
terms in x decreases by the same) that a be not less than {m— t){i — n) ; 
let then (o = (m — i){i — n), ^j,,„ becomes of the form J.a;'~" + 5<»'"~"~^ + &c., 
where A is of zero dimensions ; but this is impossible if t — ox n, for then 
J.a;'~" + &c. is a conjunctive of weight lower than the lowest-weighted 
simplified residue of the degree i — co. Hence co is not greater than i — n, 
that is (m — t) (t — n) is not greater than t — n, that is m — t cannot be greater 
than 1, that is i, when intermediate between m and n cannot be less than 
m — 1, otherwise ^^„ will vanish identically. Moreover, when i = m—l, 
(o = c—n, and i — to = n, and accordingly ^„^,„_i_^ is not merely, as we might 
know, cb priori an algebraical, but more simply a numerical multiple of (f> for 
all values of v. The same is of course true also, m being greater than n, for 
every form ^„,m_B, since this is always a conjunctive of /and ^, of which the 
former is of a degree higher than the ^ in question, so that the multiplier 
of/ in this conjunctive must be zero*. 

Art. 25. To enter into a further or more detailed examination of the 
values assumed by ^„_„ for the most general values of m, n, u, would be to 
transcend the limits I have proposed to myself in drawing up the present 
memoir. What we have established is, that to every form of ^„,i_s apper- 
taining to a value of i between and n, there is a sort of conjugate form for 
which i lies between m + n and m ; that for t = m — 1 or i = n, S-„,j_s becomes 
a numerical multiplier of ^ ; and that when i lies in the intermediate region 
between n and m — 1, ^^,i_5 vanishes for all values of v. I pause only for 
a moment to put together for the purpose of comparison the forms corre- 
sponding to b and to m + n — i. By Art. 16, making i = v-'rv, 

\h,g^^, hg^„ . . . hgj [t)^^^^ , ^f^^^ ■ ■ • V (J 

* It thus appears that if the indices m and n do not differ by at least 3 units, S^ will have an 
actual quantitative existence for all values of i between and m + n; or in other words, the 
failure in the quantitative existence of the forms 3-i only begins to show itself when this difference 
is 3 ; thus if m=n + S, &„ exists, and ^^+2 exists, but 3-„^.i = 0. 



57] of two Algebraical Functions. 471 

The conjugate form for which i =m + n— i and m — v,n — v, vv take the 
places of V, v and {m — v)(n — v), will be got by taking 






L^<r^l. Kv^ ■ ■ ■ KmJ b^+i- '?J„+3 ■ • • V(J 

which it will be perceived are identical, term for term, in the fractional 
constant factor, and differ only in the linear functions of x, which in ^^ and 
in 'A,' are complementary to one another. Our proper business is only with 
those forms for which t <n. 

Art. 26. It will presently be seen to be necessary to ascertain the 
numerical relations between ^o,i and S-^^o when i<n, and this naturally brings 
under our notice the inquiry into the numerical relations which exist between 
the entire series of forms ^„,i_„ for a given value of i, corresponding to all 
values of v between and t inclusive. 

In order to avoid a somewhat oppressive complication of symbols, I shall 
take a particular numerical example, that is m = 7, ?i = 6, t = 4, and compare 
the values of ^(,_4; %,3; ^2,2; %i; ^4,0, all of which we know to be identical 
[to a numerical factor pres] with one another and with the second simplified 
residue to / and </>, that being of the fourth degree in x ; our object in the 
subjoined investigation is to determine the numerical ratios of these several 
forms of S^ to one another. 



First, let V = 0, j^ = 4. The leading coefiScient S-0,4 is 



\VsV6 1 



\j)lV2VlV4_ 

which we know a priori (it should be observed) to be essentially an integral 
function of the h and the rj system. In this, the term containing rj^' will be 
evidently 

r.. 1 



h 1 



(A) 



the r] system to which the latter summation relates being now reduced 
to consist of i?!, %,%, jy^.T^s- In this expression, again, the coefficient of 775* 
is evidently 1. Hence, therefore, the leading coefficient in ^p^4 contains the 
term 975=*??/. 



472 On a Theory of the Syzygetic Relations 

Secondly, \Qt v = 1, v = 3. The leading coefficient in %^i becomes 



[57 









•] 






In this, the factor affecting Tji will be 












rjg being now understood to be eliminated out of the t; system included within 
the above summation. Again, in this latter sum the factor affecting t)^^ 
will be 









r 



(B) 



7^5 and 775 being now both eliminated out of the tj system. This last sum can 
of course only represent a numerical quantity. 

So in like manner, again, if v = 2, v=2, the coefficient of ijs^t?,^ in ^o_2 
will be similarly reducible to the form 






luLL 






So, again, when v = '6,v=l, the coefficient of 77/ tj^ in ^3,1 will be 



Vi 

hhh.. 






hJlshsh^^ ^ r7727;37/4 
and finally, the coefficient of 77/7;53 in ^,,4 will be 









(C) 



(D) 



(E) 



out of all which sums it is to be remembered that t/s and 776 are supposed 
excluded from appearing. All these several coefficients being numbers 
in disguise, we may determine them by giving any values at pleasure 
to the terms in the h and 77 system. 



57] 



of ttoo Algebraical Functions. 



473 



Let now 771 = hi, rj„ = h^, t), = A3, ijt = hi, then in (B) it will readily be seen 
that all the terms included within the sign of summation vanish identically, 
except the following, namely, — 






l\h„_ 



h^hjijh, 



h^h^h^h^hji-j 



14" 1 

J \jliViVs_ 



./'3 J L 



'73 

hrji^hji^h^h-j 



hjijiji^h^h 
hs 



viVsv^i ^ rv-2 

h I ' ihihhih^hsh, 



hihjiji^hjh 



[" 1 

L'7i%'74j 



'n2V3Vi\ 
A J 



[hjiji^hji^hj 



hjish^hjish-; 



•IxT- ■ 

J L'?-2'73'74. 



In each of these expressions the first factor of the numerator is identical 
in value (by reason of the equations ^1 = 771, h2 = T)„, As = 173, ^4 = 774) with 
{—f X the second factor of the denominator, and the second factor of the 
numerator with (— )" x the first factor of the denominator; hence the 
coefficient of 775^176^ in ^1,3 is —4. 

In like manner the only effective terms of ^„ ,, will be 

1 



hsh 



-1 X l'^''^' 
4J [hh. 



h.h.hj 



li^h^h^hjhj 
'V1V2'] ^ [v-2 



V1V2. 
hshs,hj^ 



''H X h'''' 



h.hJu 



h^h^h^h^hj 
h,X 






]4: 
;]4: 



hshjiji^h- 
Kh, 









h.hJir 



hihhghj 






H: 



h^hji^hji- 
h'jh 



\jl-2Vi_ 

V1V4 
h.hshhshj 



hr,hjishji 
hJh 



J L'?l'74. 



Any other term will necessaril}^ contain in the numerator a factor, whose 
symbolical representation will contain one of the quantities r]i>V2,Vs,V4> ™ ^^^ 
upper line, and one of the quantities h^, A,, h^, h,, having the same subscript 



474 On a Theory of the Syzygetic Relations [57 

index, in the lower line, and which will therefore vanish; the number of 
effective terms being evidently the number of ways in which four things 
can be combined 2 and 2 together, and the value of each term is evidently 
(_)2 2 (-1)2 15 I gQ i-jjg^^ jjjg entire value of the coefficient of 775^7;/ in ^„ 2 
is + 6. 

Precisely in the same manner, we shall find that the leading coefficient 
in ^3,1 will contain the term - 477/97/, the (- 1) resulting from the operation 
(- Xf = (-)' ^, and in "^^^^ the term + 775^77/, the + 1 resulting from the operation 
(-iy». Hence it appears that S^„,4; ^,,3; ^2; %,i; %,o are to one another 
in the ratios of 1; -4; 6; -4; 1; and so in general for any values of 
m, n, i (t being less than m and less than n) it will be found that 

will be in the ratios of the numbers 

1; (-l)™-it; (-l)2(m-2)tiZLl. (_l)3(m-3) jinlii:^^ __. (_iy(m-t)_ 

Art. 27. The method employed in the preceding investigation will 
enable us to afiSx the proper sign and numerical factor to ^^ or ^;,o, or in 
general to ^s,^-^, in order that it may represent the Bezoutian secondary 
of the degree t in x. This latter has been already identified with the 

simplified residue obtained by expanding ^ under the form of an improper 

continued fraction. For this purpose, it will be sufficient to compare a 
single term of any such ^ with the corresponding one in the Symmorphic 
Bezoutian secondary. Let us first suppose that m = n, f and (/> being of 
the same degree. A glance at the form of the Bezoutian square will show 
that if we form the Bezoutian secondary of the degree (n — i) in x, the 

coefificient of its leading term will contain the term (— ) ' ^ (0, if ; (0, i) 
as usual denoting the product of the coefficient of »"• in / by the coefficient 
of «"""'■ in ^, less the product of the coefficient of x^ in ^ by that of a;"""* 
in /; and as we suppose the first coefficients in / and ^ to be each 1, if 
we term the other coefficients last spoken of a; and «; respectively, this 
said coefficient of the leading term of the ith Bezoutian secondary will 

contain the term ( — ) ^ (a^ — a;)*, and consequently (—1) ^ «/ and 
(-yTa/. 

Now by the like reasoning to that emploj^ed in the preceding article, 
the coefficient of the leading term in ^m-i,o; that is 

X{x- h,,J (X - h,,J ...{X- h,J - L^ " ^— ^"J 



pTP K ■•■Kl' 



57] of tivo Algebraical Ftmctions. 475 

will contain the quantity "SiQiih^hs-.-Jiiy, and therefore will contain a 
term {'^{hih^ha... hi)Y, that is {—y^af, which is equal to {—fai, since 

.8-1 

{i — 1) i is always even. Hence %n-i,<i = (— )* ^ x tbe corresponding Bezoutian 
secondary. 

Art. 28. The above applies to the case where we have supposed m = n. 
When this equality does not exist we may proceed as follows. Prefix to 
^x, the first coefficient of which is still supposed to be 1, a term ecc™, where 
6 is positive and indefinitely small, and let ^x so augmented be called <1> {x). 
Then if ■jji, % ... Vn £ii'e the roots of <^x, ?7i, % ••• Vn, together with the {m — n) 

values of j-j™~"^ ■\vill be the roots of <!>(«). 



But it has already been proved that when (as here supposed) the first 
coefficient of /*• is 1, the Bezoutian secondaries to / and ^ will be identical 
with those to / and ^ respectively ; at least it has been proved that these 
latter, when 6 = 0, but the form of $ is preserved, become identical with the 
former, and consequently the same is true wheu e is taken indefinitely small. 
Now if we call the (m — n) roots of <I> which do not belong to (f>, nn+i, 
i;,j+2 . .. 77„i, and make 



we have 



where 



\K' K ■■■K~y 
^,„-M = SP (A,„ h,,... h,) U'"" ^^-^ ■ • ■ M , 

LVn+l, Vn+i---Vm} 






.■■■k:\ 

P (A„ , A,, . . . /»,,) = (^ - hq,J {X - h,,J ...{X- h,J ^h^' ul''\'^j}^^^ 
But since 77,1+1, Vn+i ■■• Vm are infinite in value, 

Hence ^,n-i, « = ( J)' ^P (Aj, , h^^ . . . h,,) 



{} 



= -]% 



Jm^i,03 



and %,i-i, = e' ■*"m-i, • 

But by what has been shown antecedently, taking account of the fact of the 



476 On a Theory of the Syzygetic Relations [57 

leading coefficient of <I> being e in place of 1, which introduces the factor e'', 
we have 

•: ^ in-i,o — \ I -Oi , 

where Bi is the Bezoutian secondary of the (m — z — l)th degree in x to /and 
^ ; but Bi has been proved = Bi, the Bezoutian secondary of the same 

.i-l 

degree to/ and <f> ; hence ^m~i,o = {-f '^ Bi. 

Art. 29. If now we return to the syzygetic equation, •?/— <^ + S- = 0, 
^ may be treated as known, having in fact been completely determined 
as a function of the roots, as well in its most general form, as also so as to 
represent the simplified residues to / and ^ in the preceding articles ; it 
remains to determine the values of t and t as functions of the roots corre- 
sponding to any allowable form of ^, but I shall confine the investigation 
to the case where S- is the lowest-weighted conjunctive or, which is the same 
thing, a simplified residue to / and ^ of any given degree in oc ; each value of 

T . r/i 

- will then represent one of the convergents to ~ when expanded under the 

form of a continued fraction. If ^ be of the tth degree in x, t is of the 
degree {n — i — 1) and t of the degree {m — i — V). This being supposed, and 
calling n — i — \=v,in— i—\=fj,,l say that t will be represented by G and 
T by r, where 

\K> K ■■■ VI 

G = (-)' 1(X- hj (X - h,) ...{X- h,J U" V' -VnJ ^ 

LV+i' V+2--- VJ 

and T is an analogous form T ; h^, lu...hm, as heretofore, being the roots 
of / and 77] , tj^ ...rjn of (f>. To fix the ideas and make the demonstration 
more immediately seizable, give m and n specific values ; thus let m = 5, 
w = 4, t = 2, so that /i=5-2-l = 2. Put S^ under the form ^,,o> so that 
^ in the case before us 



IKK. J 

Now make x = hi, then/= 0, and S- becomes 



^(h,-h,Xk-h,,) 



h„ 



?3 K Kl 

?■ K J 



67] 



that is 



of two Algebraical Functions. 



477 



Ai being kept constant in the above sum, but h^, hs, A4, h^ being partitionable 
in all the six possible ways into two groups, as into h^, h^ ; h^, h^ in the term 
above expressed. This sum is evidently identical with 



FAi h„ h 
V Vvi Vi Vs Vi^ 



h Ju 



U4AJ 



Again, (p becomes 



Hence t ■■ 



becomes 



that is P' IxSL^fl^ 

lh,h,j 

L'?l';2l?3'?4j 

[hh, 1 

^ \jllv2vsvij 






But, when x=^\, -—r^ becomes 



r/i2 hs 1 

[h 1 L^i V'- Vs Vi] 
iLh,] \h„J 



that is 



/)„ h. 






Ai hi h_ 

/12 hs 

'7i V-2 Vs V' 



h^h; 
h 



A3I 



Thus when sc = h^, t=G. In like manner, when « = /?2, or A3, or A4, or A5, 
^ always = G ; but t and (? are both functions of a; of the same, namely 
of only two, dimensions in x. Hence t is identical with G. So in general 
it may be proved, that whenever « = Aj or Aj or A3 ... or A„, t and G, which 
are each of only (m — 1 — t) dimensions in x, are equal. Hence universally 
t=G,a.s was to be shown. To find t we must avail ourselves of the sym- 
morphic, or as we may better say (it being at the opposite extremity of the 
scale of forms), the antimorphic, value, of ^ represented by S^o, „ taking care 
to preserve S- strictly identical under both forms of representation, in point 
of sign as well as quantity. That is to say, we must make 



478 On a Theory of the Syzygetic Relations [57 



hi, h, ...km 



= (-)" S (a; - vg,) (« - Vq^) •■•(«- Vq) 



.Vqv Vq2 ■■■Vq, 
Vq,+i> Vq,+2---VQn 

hi, A.2 ■■■h,n 



iVqt, Vq2 ■■•Vq,. 



where 
so that 



(1)= i,(m — i) + in (n — i), 

/ \(o _. / \mt— t-Hmi— 3nt ^ / \)nn—L . 

and consequently the same reasoning as was applied to t to prove t= G, will 
serve to show that — t = F, where 



r%' ■^h ■■■%' 

rh„ h, ...A,,n 



where 



" \%' \. ■■■% 

a> = mil — 1 — 7)11' = mn —1—m {n — t — 1) 
= mi + m— 1. 



Art. 30. I have not succeeded in throwing t and t under any other than 
the single forms for each above given, and it is remarkable that whilst 
apparently t and t admit only of this single representation, '^^ admits of the 
variety of forms included under the general symbol S-„_j_„ for a given value 
of t ; and it ought to be remarked that these forms, although the most 
perfectly symmetrical and exactly balanced representations, and for that 
reason possibly the most commodious for the ascertainment of the allotrious 
factor belonging to them respectively, by no means exhaust the almost 
infinite variety of modes by which the simplified residues, that is, the 
hekistobarytic, or if we like so to call them, the prime conjunctives, admit of 
being represented as functions of the roots of the given functions ; for if in 
Art. 16, instead of writing 



i2 = 






57] of tivo Algebraical Functions. 479 

we had made 

R = , 



where P represents any function symmetrical in respect of hq^, hg^ . . . hg , 
and also in respect of 7;, , 17^ ■■■ Vt • (the interchanges, that is to say, between 
one h and another h, or between one 7; and another t}, leaving P unaltered), 
it might be shown that the value of ^^.i' resulting from the introduction 
of this more general value of R would (as for the particular value assumed) 
always be expressible as an integral function of the roots ; and consequently, 
if P be taken of the same dimensions in the roots as the numerator of M 
previously assumed, that is vv, S-„_p would continue to be (unless indeed it 
vanish) identical (to some numerical factor pres) with the corresponding 
simplified residue. If, on the other hand, P be taken of less than vv 
dimensions in h and tj, we know d priori that S-„,;, must vanish, as otherwise 
we should have a conjunctive of a weight less than the minimum weight. 
When P is of the proper amount of weight vv, it is I think probable that 
another condition as to the distribution of the weight will be found to be 
necessary in order that ^j,_;, may not vanish, namely, that the highest power 
of any single h in P shall not exceed v, nor the highest power of any single 
7] exceed v. But as I have not had leisure to enter upon the inquiry, the 
verification or disproval of this supposed law, and more generally the evolu- 
tion of the allotrious numerical factor introduced into S^^_^ by assigning any 
particular foruL to P satisfying the necessary conditions of amount and 
distribution of weight, must be reserved, amongst other points connected 
with the theory of the remarkable forms (19) Art. 15, as a subject for future 
investigation. 

Art. 31. A property of continued fractions, which, if known, I have not 
met with in any treatise on the subject (but which has been already cursorily 
alluded to in these pages), gives rise to a remarkable property of reciprocity 
connecting t and t severally with B- in the syzygetic equation t/— id) + S- = 0. 

Let the successive convergents to the ordinary continued fraction 

1 J_ j^ _1_ 1 

qi+ qi+ qs + "' qi-i+ qi 

be called 

respectively ; it is well known that 

mi-Ji — mili_i = (—)'"' 1 ; 



480 On a Theory of the Syzygetic Relations [57 

but I believe that it has not been observed that this is only the extreme 
case of a much more general equation, namely 

where /^i, ii^-.-ixi denote respectively the denominators to the convergents 
to the continued fractions formed with the quotients taken in a reverse order, 
that is, the continued fraction 

J 1 ]_ 1 1 

qi + qi-i + 5'i_2 + ■ ■ ■ q^ + qi' 
This is easily proved when p = \; yiio is of course (as usual) to be considered 1. 
So more simply for the improper continued fraction, 

A = J_J_ 1 1 

frii qi-q,-"'qi~z-qi' 

of which the convergents are supposed to be 

m^ ' m,^'" mi_i 'mi' 
and the reverse fraction 

J^ L 1 1 

qi-qi-i'" q^-q^' 

of which the convergents are supposed to be 
^ ^ ^ 

we have the moi-e simple equation 

liVii^p — k-^mi + /Ap_i = 0. 
And it is well known, or at all events easily demonstrable, that 
li^ ^ _1 1 1^ 1 

k qi-qi-i-qi-i'" q^' 

mi_i _ J^ Ji 1^ 1 1 

m; qi - 5'j_i - qi_„_ '" q^-qi' 

Art. 32. If now we use subscript indices to denote the degree in x of the 
quantities to which they are afBxed, we have the general syzygetic equation 

KTn-.-J^ - Kt„^,_, </,„ + Z&, = 0, 

where K, a constant (which I have given the means of determining in the 
first section), being rightly assumed, Krn-.-i, iTim-i-i become the numerator 

and denominator respectively of one of the convergents to ^ , expressed as 

' See London and Edinburgh Philosophical Magazine, " On a Fundamental Theorem in the 
Theory of Continued Fractions," Vol. vi., October, 1853. [See below.] 



57] • of two Algebraical Functions. 481 

an improper continued fraction, and K^^ becomes the denominator to one of 

the convergents to ^^ , or, which is the same thing, to -~^ *. Conversely, 

J 9 

it is obvious that if we adopt as our primitive functions c/,„ and ^^-i, 

c being the value of K when t = 0, we shall obtain as the general form of 
our syzygetic equation, bearing in mind that (m - 1) now replaces n, 

ciir'T„_,_i/,„ - Z'S,^,_ii,„_i + K't, = ; 
and similarly, if we adopt as our primitive functions t„_i and c</),i, we obtain 
for our general syzygetic equation, observing that {n— 1 ) now replaces m, 

ir'^„_,_iT„_i - cK'tm-,-i<pn + K't, = ; 
so that (making abstraction of the constant factors and looking merely 
to the forms of the several functions which enter into the equations) we see 
that on the first hypothesis, namely of <,„_i being substituted for ^„, the con- 
junctives of each degree in x change places with the second conjunctive 
factors, that is the original multipliers of ^ of the same degree in x, and 
vice versa; and in the second hypothesis, where t„_i takes the place of /"^, 
the conjunctives of each degree in x change places with the first conjunctive 
factors, that is the original multipliers of / of the same degree in x, and 
vice versa ; i,^i and t„_i being respectively multipliers of <^ and /, such that 
the difference of the respective products is independent of x. These results 
ought to be capable of being verified by aid of our general formulae for t, t, ^, 
and as this verification will serve to exhibit in a clearer light the nature 
of the reciprocity between the conjunctives and the conjunctive factors, 
it may be not uninteresting to set it out. 

Art. 33. As usual, let h^, h„... hm be the roots oi fx, and 171, %...»;„ 
the roots of <j)x; the last conjunctive factor to </>, which is of the degree 
(m — 1) in X, will be represented, neglecting powers of (— ), by tm~i, where 



tm-i = 2 (» - hg^) (x - /(-jj ...(«- Ag,„_j) 



Vn 



-Qi ' % • • • "?m-l_ 

If now we for greater simplicity make C-i = t (x), and call the roots of t, 
v'i> v's ••• v'm-i, any such quantity as 

(^?m - K) e*?™ - K) ■ ■ ■ (Km - Km-,) 

■ -7? 1 

* Since i is always supposed less than n (n being the degree of the lower degreed of the two 
functions / and 0), the fact of the last quotient to ~J being wanting to ^^' will not affect the 

/ 

accuracy of the statement in the text above, since this latter will contain as many quotients as 
can in any case be required for expressing &i . 

s. 31 



482 On a Theory of the Syzygetic Relations [57 

R denoting a constant independent of the root hq^ selected, in fact the 
resultant of the two functions fa and ^x, that is to say, 

4>{K)<^{K)4>{K)...4>{hr,). 

But by our general formulas the simplified residue to fa and t {x) of the 
tth degree in x will be represented by 

V'u V'i •••Vm-i 



^;,„ = t{x- \) {x - hq^) ...{x- hj j L_^^^ — t^^. h 
therefore 






Asi+l' "?i+2 •■• "'Sm 



Sr', = i?™-'-ii„ 



the relation which was to be obtained. So conversely, in precisely the same 
manner, calling i\ the conjunctive factor of the degree i m. x to t (x) in 
the syzygetic equation which connects fa and t (x) with a corresponding 
simplified residue, we have 

Hy /ig, ... V 1 . 

«-?, > «gj ■•■• "'q, 

L"Si+i' Si +2 ••■ "SmJ 

= ii;--2 (^ - A J (^. - A,,) ...{X- h,) '^ ^Y:'^ "^ ^^--^ • • ■ "^ ^^^-^ 

"-?!' '^92 ••• % 

L"?i+1' %+2 ••• "'ImJ 

= R'-'X, 

the conjugate equation to the one previously obtained*. 

And evidently the same reasoning serves to establish the reciprocity, 
or rather reciprocal convertibility, between the ^ series and the t series, 
when in lieu of the original primitives fa and <j}X we take as our 
primitives t {x) and (^a;, t (x) being the function which satisfies the 
equation 

T (x)fa - t (x) (px + ^ = 0. 

* M. Hermite, by a peculiar method, first discovered one of these two conjugate relations of 
reciprocity, applicable to the case of Sturm's theorem, where 0a; =f'x, and I ara indebted to him 
for bringing the subject under my notice. 



57] of two Algebraical Functions. 483 

Art. 34. It may be remarked that if m = m — 1, the last syzygetic 
equation being thus t,n_j,<p„i_i — Tn-ifm — ^o = 0> when i,„_i and /^ are taken 
as the primitives, the corresponding equation will be of the form 

* »»— l*m— 1 "r m—zjm + ^ o ^ " ; 

these two equations must therefore be identical, and consequently i'„i_i = <pm-\ 
(to a numerical factor prh), so that tm-i and (^„i_i are reciprocal forms ; this 
is also obvious from the consideration that <',„_! must, by the general law 
of reciprocity (established above), be a residue to {fm, <f>m-\), which the 
latter function itself may be considered to be. Or the same thing is obvious 
directly, by writing 

and then making 

V'Sm ~ "gj ■ ■ ■ \"'qm ~ "■Qm-l) 

— rfm-iS(r — h ^ (r-h \ fjjhml 



or finally, 

as was to be shown. 



t', = R""-"- , 



Section III. 

On the application of the Theorems in the preceding Section to the expression 
in terms of the roots of any primitive function of Sturm's auxiliary 
functions,' and the other functions which connect these with the primitive 
function and its first differential derivative. 

Art. 35. The formulae in the preceding Section had reference to the case 
of two absolutely independent functions and their respective systems of roots : 
when the functions become so related that the roots of the one system 
become explicitly or implicitly functions of the roots of the other system, 
the formulae will become expressible in terms of these latter alone, and in 
some cases the terms (of which the sum is always essentially integral) will 
become separately and individually representable under an integral form. 
Such, as I shall proceed to show, is the case for two functions, of which one 

31—2 



484 On a Theory of the Syzygetic Relations [57 

is the differential derivative of the other. When / and ^ are thus related, 

so that <i> = -T- , calling as before K,h^... A,„ the roots of/ and Vi,V"---- Vm-i 
^ ax 

the roots of </>, we shall have in general 

L'7l. V^-'-Vm-lJ 

Consequently 









Us.. 



Hence 



hi' 






1?!, V2---Vm-y 
[t/i, ija . . . ??,ft_i J 

'''9i+2 

"■o.j_, I '^O.-j.. • • • "o,., 



J 
J 

J- 



"■9i+2 

'72 



^9» ' 
• Vm-1 






Ao,^„ . . . A„ 



rvi' ^9i+2---^ 



"Qi+S • • ■ 9m J L 9i+l ' 9i+3 • 



h- 1 

L"'9i+I' "9i+2 • • ■ "'9m-l J 



the f denoting the operation of taking the product of the squares of the 
differences of the quantities which this symbol governs. Hence the Bezoutian 
secondary to/ and/' of the {m — i— l)th degree in x, namely 



(-y-T-iix- h,,,,) (x - h,,j ...(X- h,j 



\K' K---K 1 

\_Vi, '?2 •■• '?m-ij 

\K' K •■■K'Y 



becomes 



(-)*<-" 2?(A,„ h,^...h^}{X-h^,^;){X-h,,^,)... {X-hgJ 

= 25" (^9.. K-" K) (« - ^9i+i) (^ - ^9i+2) ■■■{X- hgj, 



57] of two Algebraical Functions. 485 

since {—f ''~" = 1 ; this gives the well-known formulse (enunciated * by me 
in the London and Edinburgh Philosophical Magazine for 1839) for expressing 
M. Sturm's auxiliary functions in terms of the roots of the primitive, and 
which I therein stated were immediately deducible from the general formulae 
(also enunciated in the same paper) applicable to any two functions. These 
more general formulae appear to have completely escaped the notice of 
M. Sturm and others, who have used the special formulse applicable to 
the case of one function becoming the first differential derivative of the 
other. 

Art. 36. In precisely the same manner, if we form as usual the ordinary 
syzygetic equation 

tf'x - t/c + S^ = 0, 

we may find the different values of t given by the complementary formulse ; 
and using ti to denote the multiplier of the degree i in x, that is appertaining 
to the residue of the degree {m — i—\) va. x, we have 

ti = tji r^'' I -^7^ (^-fe,.)(^-/iJ...(^-A,,) 

= S^ (hg^, hq, . . . hg.) (W - hg) {X - hg^) ...(«- /(,;). 

Art. 37. Thus, if we make i=m — l, 

//a; = i™_, = Sf (/tg,, hg,^... h.g^_^) {x - hg) {x - hg) ...{x- hg^_^). 

It is evident from the form oifi'x that it possesses relative tofx, the same 
property as _/'«, I mean the property that when x is indefinitely near to a real 
root otfx, and is passing from the inferior to the superior side of such root, 

'^~- like At- will pass from being negative to being positive, or in other 
jx jx 

words, f-[x and f'x have always the same sign in the immediate vicinity 

to a real root of fx. Hence it follows that //« might be used instead 

of f'x, to produce, by the Sturmian process of common measure, a series 

of auxiliary functions, which with fx and f^x would form a rhizoristic series, 

that is a series for determining (as in the manner of M. Sturm's ordinary 

auxiliaries) the number of real roots of fx comprised within given limits. 

The rhizoristic series generated by this process will, it is easily seen, be (to a 

constant factor pres) the denominators (reckoning + 1 as the denominator 

f'x 
in the zero place) of the successive convergents to"^^ thrown under the form 

[* p. 45 above.] 



486 On a Theory of the Syzygetic Relations [57 

of a continued fraction . . . ; M. Sturm's own rhizoristic 

gi - ^2 - qn-i - qn 

series, on the contrary, will be (to a constant factor pres) the denominators 
of the convergents to the inverse fraction '^4— , which will be of the form 

K ( . . . 1 ; accordingly these two rhizoristic series will be 

\qn - gn-i - qi- qJ 

equivalent as regards the number of changes and of combinations of sign 
(afforded by each) corresponding to any given value of x, of which of course 
the g's are linear functions. This result agrees with what has been demon- 
strated by me* by a more general method (in the Lojidon and Edinburgh 
Philosophical Magazine, June and July 1858), where it has been proved, 
by means of a very simple theorem of determinants, that the two series 

qi' qi- q^' qi- q-.- q^ '" qi- q2- q^'" qn' 



and 



111 111 1111 



qn qn qn—i qn qn—i qn—2 qn qn—1 qn—2 qi 

always contain (for real values of ^i, ^a, ^s •■■ qn) the same number of positive 
and negative signs. 

Art. 38. Having now determined the general values of ^ and t in 

the equation tf'x — rfx + ^ = as explicit integral functions of the roots 

of fx, the more difficult task remains to assign to t its value similarly 

expressed. This cannot readily be effected by means of substitutions in the 

general formulae, the method we adopted for finding t and S^ ; but all the 

other quantities except t in the syzygetic equation being integral functions 

of the roots, it is evident that t also must be an integral function of the 

tf'x + ^ 
same, and to obtain it we may use the expression t = - — -^ — . 

To obtain the general form of t by direct calculation from this formula 
would however be found to be impracticable ; the mode I adopt therefore 
to discover the general expression for t corresponding to different values 
of ^, is to ascertain its value on the hypothesis of particular relations 
existing between the roots oi fx, and then from the particular values of t 
thus obtained to infer demonstratively its general form, as will be seen 
below. The demonstration of t is unavoidably somewhat long, t being in 
fact represented by a double sum of partial symmetrical functions. 

Using the subscript indices of each function as the syzygetic equation 
to denote its degree in x, we have in general 

tm^i-J'x - Tm-i-2fx + ^£ = 0, 
[* See below pp. 616 and 621.] 



57] of two Algebraical Functions. 487 

where if we make 

Ai — a; = ^1 , h^ — X = kz . • . hm — so = km, 
so that 

hi — h^ = ki — k^, 
and therefore 

^{hg^, he, ... /tflp) = ?(&«,, kg.^ ...kg^), 

we have in effect found 

a-j = ZkgJCq^ ... kq. ^{kq.^^, kq.^,, . . . kqj) 

and 

*m-i-l = ± 2feg, kq^ . . . kq^_^_^ ^ (kq^ , kq^ . . . Kg„_,._j) ; 

we have also /'(«;)= {—)^~^ Ikqkq^ ... kq^^_^. 

Let us commence with the case where i = ; we have then 
'^,= ^{h,h...km), 

tm-l = Zkqkq^ . . . kq^_^ K\kqo kq,^ . . . kq^_j) ; 

we have thus 

\ / ''"m— 2 ""i ^2 . • • ""m ^^ b \"'i 3 "-2 . . . f^m) 

— Zkq^kq^ . . . kq^_j X jL^Kq^kq^ . . . kq^_^ ^{kq^, kq^ . . . kq^J. 

It may easily be verified that the negative sign interposed between the two 
parts of the right-hand member of the equation has been correctly taken, for 

^(yfci, h^... k,n) contains a term A;i-'™-^' i-^^""--' ... ¥m-ik"'rn-u 
"^kqjcq^ ... kq,^^ contains a term k-^k^ ... k^-^km-i, 
and 

Ikq^ kq^... kq^_^ f {kq^ , kq^... kq^__^) Contains a term k^""-^ k^""^' . . . k?m^^ k'^^j , 

and thus the term V'™"" ^2^""~^' •■• ^m-a^m-i, which does not contain 
kik2...km, will (as it ought to do) disappear from the right-hand side of 
the equation. 

Now suppose 

then 

^{k„ k,...km) = 0, 
and also 

b (kq^, kq^ . . . kq^^_J = 0, 

except when one or the other of the two disjunctive equations 

qi, q^, ?3... qm^i^l, 3, i ...m, 

qi, 9'2, gs . . ■ S'm-i = 2, 3, 4...m, 

is satisfied (by a disjunctive equation, meaning an equation which affirms 
the equality of one set of quantities with another set the same in number, 
each with each, but in some unassigned order). 



,4h. 



488 On a Theory of the Syzygetic Relations [57 

Hence 

Hence when ki = k„, (— )'"t,„_2 becomes 
2 

j^ '^kgj^qi ■ ■ ■ ^'gm-l ? (^1> ^3 • • • k,n), 

that is 

2^{k„ h...k^) {k^lk,.,kr,... A;,,„_, + ^kk, ... k,n}, 

the S referring to ?'3, j-j ... ?',„ supposed to be disjunctively equal to 3, 4 ... m. 

Now Tm_2 is of (?» — 2) dimensions in a;, and whenever more than one 
equality exists between the ^'s, &» and t^-i both vanish (in fact every term 

in each vanishes separately), and therefore T,n^2> which = - ° , ™ V — > 

"'I "'2 • • • "'m 

will vanish. 

Hence (— )™ t„i_2 must be always of the form 

■^ denoting some integral function of (m — 2) dimensions in respect of the 
system of quantities kq^, kg, ... kq„^. The result above obtained enables us 
to assign the value of 

^(^1, h ...km, k), 
when ki = k, namely 

kiX(kr^, kr^ ... kr^_^) + 2k.,ki . . . k„,. 

Now for a moment suppose, selecting {m — \) terms k^, k^, k^... km out 
of the in terms of the k series, that 

Q, (L\, ks, h... km, h) = ky^-^ - k^^-^S, (k, ,k,... k^) + h^'S., {k, ,k,...km} 

± ...+ LSm^, (k^, k^... km) ± 2Sm-2 (ki, h... km), 

where S-^ means that the quantities which it governs are to be simply added 
together, S„ denotes that their binary, S^ that their ternary, and in general 
Sr that their ?'-ary products are to be added together. 

When ki = k2, fi becomes 

^ m-2 _ ^ m-3 {^^ 4. S^ (^^ ^ ^,^ . . _ ^^^)} + ^ m-4 \J^^ gy {h ,h... km) +S^{h,K... km)] 

- k^-' {hS^ (h,k,... km) + S, (k, ,k,... km)} ±... 

± h [hSm-, {h,h... km) + S,„._, {k,, k,... km)} ± 2Sm-2 {h, h--- K), 
which evidently equals 

± {2<Sf„^_2 (^3. h--- km) + kySm-, (k„ k,... km)}, 
that is ±{k,t(kr„kr^...kr,„^^) + 2k3k,...km}. 



57] of two Algebraical Functions. 489 

Hence when ^i = k., "•¥ = il, and 

(-)» T^, = X^{h,^, hg, . . . hg^_^) X fl (A,-,,, ^,^ . . . ^"^,^.„ A;,^) ; 

and so in like manner, when k^ is equal to any one of the {m — 1) quantities 
/cj, k^... km, the form of rm-2 above written will have beeu correctly assumed. 
But Tm_2 maybe treated as a function of (m — 2) dimensions in k^, and 
consequently any form of (m — 2) dimensions in k^, which fits it for (m — 1) 
different values of k^, must be its general form, and accordingly we have 
universally, 

(-r r,n-. = 2? (hg, , h,^ . . . hg,,J X {{X - /l,„)-- 

± 2*Sm_2 (x-hg^,x-hg„...x- hg^_^)}. 

Art. 39. With a view to better paving our way to the general form of 
T for all values of i, let us pass over the case of i = 1 and go at once to the 
equation 

tm-,f'«! - -Tm-Jx + ^2 = 0; 

and to better fix our ideas let ?h = 7, so that the equation becomes 

tif'x — Tifx + S-2 = ; 

we have then, preserving the same relation as before, that is, using h to 
denote any root oifx, and k to denote /;. — x, the equation 

+ Ki K^ Kg Ki ft's ^5 Kj Tg ^ ^kg^ kq^ ^ \kg^ kg^ kq^ kq^ kq^) 

— -kqjiqjiq^kq^kq^kq^ X S [kg.kg^kgJCg^ 's\kq,kqikq^kgj\ ) 

now Tj will vanish whenever more than three relations of equality exist 
between the k's., for then each term in both of the two sums in the right-hand 
member of the equation above written will separately vanish ; and of course 
three relations of equality between the same are sufficient to make all the 
terms in the first of these sums vanish. This relationship between the 
different k's, corresponding to a multiplicity 3 may arise in different ways ; 
the multiplicity 3 may be divided into 3 units corresponding to 3 pairs of 
equal roots, or into 2 and 1 corresponding one set of 3 equal roots, and a 
second set of 2 equal roots, or may be taken en bloc, which corresponds to 
the case of one set of 4 equal roots. I shall make the first of these supposi- 
tions, which will sufficiently well answer our purpose in the case before us. 

Thus I shall suppose 

Hj-^ ^— rC^ J n/g -— iCq j iCq ^ A^g J 

then, as above remarked, 



490 On a Theory of the Syzygetic Relations [57 

for all values of ^3, q^, q^, q^, q^, and therefore 

also '^kqjcqjcqjcqjcq^kg^ becomes 

and ^ (kqkqjcqjcq^ vanishes, except for the cases where q^, q^, q^, qt represent 
respectively, g'l the index 1 or 4, q^ the index 2 or 5, g'3 the index 3 or 6, and 
^4 the index 7. 

xlence ^"'Si"'32 Sa Ji 5 \ 31 32"'53"'9i) ^^ ICik2K3Kj ^yKiknk^K';), 

and consequently T3 becomes 

+ 8f (k-^lc^hJcT) X {^iAjjAs + 2^7 (^1^2 + ^1^3 + ^2^3)}- 
Hence we are able to predict that the general expression for our t in the 
case before us will be 

T3 = + Zi(^ {kqkqJCq^kq^) 

X {(V + V + V) - (V + h! + V) (^<r. + K + K + K) 

+ \kq^ + kq^ + kqj {kqJCq^ + kq^kq^ + kqJCq^ + kqJCq^ + %% + kqJCq^ 
— 4t{kq^ kq^ kq^ + A;g_ Ajj &j, + kq^ kq^ kq^ + kq^ kq^ kq^^ . 

For in the first place, the fact that the t vanishes when more than three 
relations of equality exist between the k's, proves that we may assume t^ 
of the form 

the semicolon (;) separating the k's into two groups, in respect of each of 
which severally ^ is a symmetrical form. But if in the expression last 
above written for T3 we make 

rCj ^ A/4 , fCt^ ^ iC^ , 7^3 ^ A^g , 

it becomes 

+ 8^ {k,k,k,k,) X {(k,' + ki + ^3^ - {h"^ + ki + k,') (k, + k2 + h + k,) 
+ (ki + k2+ ks) (kik^ + k^ks + k^k^ + k-^kj + k^kj + kik,) 

Now in general if 

o-r = a/ +a/ + a/ + ...+ at"", 

and Sr = 'Z {a^a^as . . . ar), 

then a-r - a-r-A + o",— 2^2 ± . . . ± r iS, = 0. 

Consequently the sum of the terms constituting the second factor in the 
above expression 

= (3 - 4) kik^k^ + (2 - 4) k, (k^k^ + ^1^3 + hk,). 



57] of two Algebraical Functions. 491 

Hence the above expression becomes 

+ S^ik-^k^k^kj) [k-^Kks + 2 {k^k^ + ^"i^s + k^k^ A;^}. 
Thus, then, whenever k^, k^, h are respective!}' equal to any three of the 
quantities ki,ks,ks,kj, which may take place in twenty-four different ways 
(twenty-four being the number of permutations of four things), our r^ will 
have been correctly assumed ; but ^{kqk^^kqjcq^ being replaceable by 
^ {hqjiqjiqjiq^), the T3 may be treated as a cubic function in ^i, k,^, k^, and 
arranged according to the powers of A;,, k^, k^ will contain only twenty terms ; 
hence, since the assumed form is verified for more than twenty, that is, for 
twenty-four values of /ij, h^, h^, it follows that the assumed form is universally 
identical with the form of t, which was to be determined. 

Art. 40. Now, again, in order to facilitate the conception of the general 
proof, let us suppose /"« to be of only five dimensions in x, i still remaining 3: 
it will no longer be possible when we suppose a multiplicity three to prevail 
among the roots, to conceive this multiplicity to be distributed into three 
parts, for that would require the existence of three pairs of roots, there 
being only five. But we may, if we please, make Ai=/i2 = ^3, and h,i = h^, 
or else h^=hi = hi='hi, or in any other mode conceive the multiplicity to be 
divided into two parts, 2 and 1 respectively, or to be taken collectively 
en bloc. As a mode of proceeding the more remote from that last employed, 
I shall choose the latter supposition. Then we obtain (t now becoming 
T5_2_2, that is Ti) 

lilk^K^kik^T-l^ + ^kq^kqJCq^kq^ X ^kq^kq^ i {kq^kqj, 

and ^{kqkq^ will vanish, except in the case where q^ represents the indices 
1 or 2 or 3 or 4, and q^ the index 5 ; also 

Hence our equation becomes 

h'k.T = + (V + 4!k,^h) 4>k,k, ^{hh), 
and T becomes — i^{kik^) {k^ + 4A;s). 

If, now, we assume for the general value of t in the case before us 

T = t^ikq^kq;) {{kq^ + kq^ + kqj - 4 (kq^ + kq^)}, 

when ^1 = ^2 = ^3 = ^4, T becomes 

±4^^{hh){3k,-(^h + h)}, 
that is + 4,^ (kiks) (h + ^h)- 

Hence then for the two systems of values of Aj, h^, h^, namely 

hi = hA iJh = h 

hn = hA or -j ^2 = /i5 

hs = hj m, = hs, 



492 On a Theory of the Syzygetic Relations [57 

the form of t will have been correctly assumed. But since the derived form 
is a linear function of h^, h^, h^, this is not enough to identify the assumed 
with the general form, since for such verification four systems of values must 
be taken, four being the number of terms in a function of three variables 
of the first degree. If, however, we had adopted a separation of the multi- 
plicity three into two parts, and had started with supposing ^1=^2 = ^3, 
]c^ = Ic^^ we should have found that t would have become 

Moreover, when these equalities subsist, 

kik^k^ki + Ai^'a^'s^'s + k^k^kik^ + k-jf^kjci + kji-jkik^ 

becomes ^k^k^ + Wc-ch^, and the common factor k^-k^ disappears in the course 
of the operations for finding t, and eventually we have to show (in order to 
support the universality of the previously assumed form for t) that 

becomes — ikq^ — Zk^ when 

kq, = kq^ = kq^ = k^ , 

anci A/fi ^^ A/fT ^ /C5 , 

which is evidently true. Hence then t will have been correctly assumed for 
the following cases, 

A/j ^ A/2 ^— rC^ ^ /ig 
rC-^ ^ AiQ ^ 'tg ^~ A.'^ j 

and also for the cases 

^'i = /b'a = A-'s and k^ = k^ 

ki = k^ = ks and k.^ = A4 

k„ = A-j = /I's and k\ = k^ 

ki = k2 = ki and k\ = k^ 
ki = ks = ki and L = k^ 

k^ = h = ^^4 and ki = k^ 

that is, for eight cases in all, whereas four only would have sufficed. Hence, 
ex abundantid demonstrationis, the form assumed for Tj is in the case before 
us the general form. 

Art. 41. We may now easily write down the general form which t 
assumes for all values of i and prove its correctness. If the roots be 

/ii. A.,, /is ... hill, 

and *m-i-i /'« - Tm-i-'.fa + ^; = 0, 



57] of tivo Algebraical Ftmctions. 493 

we shall have 

+ (-)'»-'■-=' aA.-i-s + (-)™--= (a, + 1) S,„_i_J}, 
where ar denotes in general the sum of the rth powers of the (t + 1) quantities 

and jSy denotes in general the sum of the products of the complementary 
{m — i — 1) quantities 

{x - hj, {x - hj ...(x- /?„,„_._,) 
combined r and r together. It will of course also be understood that 
0-0 = 1 + 1, so that cTo + 1 = I + 2. 

Art. 42. To prove the correctness of this general determination of the 
form of T„i_i_2, let us suppose in general that i + 1 relations of equality 
spring up between the m quantities ki, k^-.-km; we shall then easily 
obtain (N representing a certain numerical multiplier) 

ki,k^...k,n_i_i being what the k system becomes when repetitions are 
excluded, and being respectively supposed to occur /^i, fi2 ■■■ f^m^i-i times 
respectively, so that 

fj.i + fj'2+ ■■■ + A'm-t-i = ■»« ; 
the fractional part of the right-hand member of the equation immediately 
above written will be readily seen to be equivalent to 

^f'-em-i-i "-ei^'sj • • • "'em-i-2- 
To establish the correctness of the assumed form, we must be able, as in 
the particular cases previously selected, to prove two things ; the one, and 
the more difficult thing to be proved is, that when the series of distinct 
quantities ki, Ajj, k^ ...km become converted into fii groups of k^ ; fi^ groups of 
Ajj, ... /Mm-i-i groups of A;„i_i_i, then that 

^H-Sike-ikeskBi ■■• %„-;_!> 

or in other terms 

1 

m—i-l 

becomes identical with 

0-^_i_2 - O-jn-i-s S^ ± &C. + (- l)'«-'-2 (o-„ + 1) Sm-i-2. 

The other step to be made, and with which I shall commence, consists 
in showing that the number of terms in the expression last above written, 
considered as a function of (m — i — 2)th degree of (i + 1) variables, is never 
greater than the entire number of ways in which (i+ 1) quantities out of m 
quantities may be equated to the remaining (m — i— 1) quantities, namely 
each of the first set respectively to all the same, or all different, or some the 



494 On a Theory of the Syzygetic Relations [57 

same and some different; in short, in any manner each of the i+ 1 quantities 

with some one or another (without restriction against repetitions) of the 

TO — i— 1 remaining quantities. This latter number being in fact the 

number of ways in which {m — i-l) quantities may be combined (i+1) 

together with repetitions admissible, by a well-known arithmetical theorem, 

(t + 1) (i + 2) (m — 2) 
is (m-i- iy+', and the first number is ^^ — :r^ — . . „, , which is 

^ 1 . A ... (m — I — A) 

always less than the other. It remains then only to prove the remaining 

step of the demonstration*. 

Art. 43. To fix the ideas let m = 10, i= 5, and consider the expression 

(k,' + h' + h' + h' + k,' + kj) - {k,' + k,' + V + ^g" + V + ho') {h + k^^-k, + k,) 

+ (k^ + ks + k7 + ks + ks + ho) (hh +kik3 + hh + k^k, + hK + ^3^4) 

— 7 {kihks + k^hh + hhki + hk^kt). 

Now suppose the six quantities ^5, k^, k,, kg, k^, h^ to become respectively 
equal each to some one or another of the four quantities h, kz, h, h, as for 
instance, I shall suppose 

fC^ = fvQ ^ fC^ ^ Kl 
Kid ^ ft'3. 

Then /^i = 4, /is =3, /a3=2, /i4=l> 

and the formula of Art. 41 becomes 

(3^1= + 2k^' + h') - (3AiV+ 2k^^ + h") (h + h + h + h) 

+ (3^1 + 2^2 + ^3) (^1^2 + ^1^3 + ^1^4 + ^2^3 + KK + ^"3^4) 

= 3 [{A;i= - yfci^ {h + h + h) + h} + h {{hh + hh + k^k^ +ki{k„^+ h + h)]] 
+ 2 {^2= - ki (h + ks + h) 4 L} + L [{hka + kjki + ksh) + {hh + k,) + h] 

+ [k^^ - h' (h + h + h) + h] + h [{hh + hh + h h) + ^3 (*i + ^2 + ^4)} 

= -kihhkAP + T^ + ^ + ^' 

I A/J Ki^ A/3 A/4 

* If this first step of the demonstration appear unsatisfactory or subject to doubt, it may be 
dispensed with, and the result obtained in the succeeding article (the demonstration of which is 
wholly unexceptionable) being assumed, it may be proved that the formula there obtained on a 
particular hypothesis must be universally true, in precisely the same way and by aid of the same 
Lemma in and by aid of which the formula obtained in the Supplement to this section for the 

f'x 
simplified quotients to -z— upon a like particular hypothesis is shown to be of universal application, 

that is, by showing that otherwise a function of 2i - 1 variables would contain a function of 2i 
variables as a factor. 



57J of two Algebraical Functions. 495 

In the above investigation the quantities which with their repetitions 
make up the k's system, are ^4, Ajj, k^, k^, appearing respectively 1, 2, 3, 4 
times, that is to say repeated 0, 1, 2, 3 times ; 7 is one more than the sum 
of the repetitions + 1+2 + 3, and the numbers 1, 2, 3, 4 arise from sub- 
tracting from 7 the sums 1 + 2 + 3; + 2 + 3; + 1 + 3; + 1 + 2; respec- 
tively, so that the remainders 1, 2, 3, 4 denote respectively one more than 
the number of repetitions of ^4, ^1, k^, k,, that is, are the number of appear- 
ances of ki, ki, k^, k,; and thus with a slight degree of attention to the 
preceding process the reader may easily satisfy himself that the preceding 
demonstration (although not so expressed) is in essence universal, and the 
form of T as an explicit function of x and of the roots of fx is thus com- 
pletely established for all values of m and of i. 



Supplement to Section III. 

On the Quotients resulting from the process of continuous division ordinarily 
applied to two Algebraical Functions in order to determjine their greatest 
Common Measure. 

Art. (a)*. We have now succeeded in exhibiting the forms of the 

f'x 
numerators and denominators of -^ developed into a continued fraction in 

terms of the differences of the roots and factors oi fx. It remains to exhibit 
the quotients themselves of this continued fraction under a similar form. 

Lemma. An equation being supposed of an arbitrary degree n, there 
exists no function of n and of less than 2i of the coefficients'^, which vanishes 
for all values of n whenever the n roots reduce in any manner to i distinct 
groups of equal roots ; or in other words, any function of n and the first 2i— 1 
coefficients of an equation of the nth degree, which vanishes for all values of n 
in every case where the roots retain only i distinct names, must be identically 
zero. 

To render the statement of the proof more simple, let i be taken equal 
to 3. And let the roots be supposed to reduce to p roots a, q roots b, and 

* The articles in this and subsequent sections to which Latin, Greek and Hebrew letters are 
prefixed, although in strict connexion with the context, are supplementary in the sense of 
having been supplied since the date when the paper was presented for reading to the Eoyal 
Society. All the articles marked with numbers (from 1 to 72), and the Introduction, appeared 
in the memoir as originally presented to the Society, June 16, 1853. 

t In the proposition thus enunciated the coefficient of the highest power of x is supposed to 
be a numerical quantity. 



496 



On a Theory of the Syzygetic Relations 



[67 



r roots c. And let Sr in general denote the sum of the rth powers of the 
roots. Then we have evidently 

p + q + r = So, 

pa +g& + re = Si, 

pa?- + qlfi + rd^ = s^, 



pa^ -{■ 



■ + rc^ = s-i, 



1, 1 

h, c 
h\ c 



= 0; 



pa* + qh'^ + re'' = s^, 

&c. &c., acZ infinitum. 

Eliminating p, q, r between the first, second, third and fourth equations, 
we obtain 



= 0. 



a', b^, &, S3 

In like manner eliminating ap, bq, cr between the second, third, fourth and 
fifth equations, we have 

1, 1, 1, Si 

a, b, c, S2 

a', b", c^, S3 

a', 6^ c^. Si 
and so in general we have for all values of e, 

1, 1, 1, s, 

a, b, c, Se+i 

a^, b-, &, Se+2 

a^ &^ C^ Se+3 

whence it may immediately be deduced, that, upon the given supposition of 
there being only three groups of distinct roots, we must have the following 
infinite system of coexisting equations satisfied, namely, 

s^t + SjM + s^v + S3W = say L„ = 0, 

S],t + SsW + S3V + SiW = „ Xi = 0, 

s^t + SzU + SiV + S5W = „ L„ = Q, 

Sst + SiU + S^V + SeW = „ £3=0, 

sj + s^u + s^v + SjW = „ -^4 = 0, 
&c. &c. &c. &c. ; 



= 0: 



57] of ttoo Algebraical Functions. 497 

and conversely, when this infinite system of equations is satisfied the roots 
must reduce themselves to three groups of equal roots. 

Let now </> be any function of s^, Si, s^ ... which vanishes when this is 
the case. Then <^ must necessarily contain as a factor some derivee of the 
infinite system of equations above written, that is, some function of Sq, s,, s^... 
which vanishes when these equations are satisfied, that is, some conjunctive 
of the quantities L„, L^, L^, L^...; but it is obviously impossible in any such 
conjunctive to exclude s^ from appearing, unless by introducing some other s 
with an index higher than 6, and consequently ^ cannot be merely a function 
of So, S], S2, S3, S4, Sb, nor consequently of n and the first five coefficients ; or if 
such, it is identically zero. And so in general any function of n and only 2t — 1 
of the coefficients which vanishes when the roots reduce to i groups of equal 
roots, must be identically zero ; as was to be proved. 

Art. (6). It ought to be observed that the preceding reasoning depends 
essentially upon the circumstance of n being left arbitrary. If n were given 
the proposition would no longer be true. In fact, on that supposition, the 
n roots reducing to i distinct roots would imply the existence oi n — i 
conditions between the n roots; and consequently n—i independent equations 
would subsist between the n coefficients, and functions could be formed of i 
only of the coefficients, which would satisfy the prescribed condition of 
vanishing when the roots resolved themselves into i groups of distinct 
identities. 

Art. (c). Let D^^^r^.^n be used in general to denote the determinant 

Sti t ^ri+i I 5^1+2 • ■ • ^ri+i— 1 
Sr.2 > Sj-j+l > *r2+2 • • ■ ^rj+i— 1 



■p S/j+l) Sri+2 ■ • • ^n+i—l 

then the simplified ith Sturmian residue Ri may be expressed under the form 

n . »,n— i— 1 _ 7) . , ^n—i-2 i J) . ~n-i-s 4. 7) . 

-^1,2,3. ..l^ ■'■^2,3... l+l"^ ' ■'-^3,i...l+2'^ ••• X J-'n-l,?^— I— 1...JI) 

which is easily identifiable with the known expression for such residue. 

Now obviously the necessary and sufficient condition in order that the n 
roots may consist of only repetitions of i distinct roots is, that Ri shall be 
identically zero, that is to say, we must have 

-^1,2... i ^^ ^) -^^2,3.. .1+1 ^ " •■■ J-'n—i,n—i—i...n, ^^ '-'• 

But the reasoning of the preceding article shows that although these equa- 
tions are necessary and sufficient, they are but a selected system of equations 
of an infinite number of similar equations which subsist*, and that, in fact, 

* But quare whether any other sufficient system can be found of equations so few in number 
as this system. 

s. 32 



498 On a Theory of the Syzygetic Relations [57 

■whatever be the value of n, we may take ri, r-a-.-rj perfectly arbitrary and 
as great as we please, and the equation 

must exist by virtue of the existence of the n — i equations last above written. 

Art. {d). I now return to the question of expressing the successive 

quotients of "^ as functions of the differences of the roots and factors ; that 

they must be capable of being so expressed is an obvious consequence of the 
fact that the numerators and denominators of the convergents have been 
put under that form, since, if 

are any three consecutive convergents of the continued fraction 
_1 1_ JL^ 

we must have 

It would not, however, be easy to perform the multiplications indicated in 
the above equation, so as to obtain Qi under its reduced form as a linear 
function of x. I proceed therefore to find Qi constructively in the following 
manner. 

Let Ri-2, Ri-i, Ri be three consecutive residues, f'x counting as the 

residue in the zero place, then Qi = —^ \ and is of the form ^00 + —. 

Now in general if we denote the n roots of fx, where the coefficient 
of a;" is supposed unity, by h^Ju-.-hn, and if we use Zi to denote 
S?(V, /'e2---^ei)*> with the convention that Zi = n, .Zo = l, we have, employ- 
ing {i) to denote ^ {(— 1)' + 1}, 



_ Z-i-iZ\_i...Z- ii^+i 

Z"i_l^-j_3 ... Z 



Bi-. = ;--;-'--Jf^' 2 {?(/^e.> he, . . . he) {x - h^J (x - Jh.J ...{x- hj], 



* f it will be remembered is the symbol of the operation of taking the product of the squares 
of the differences of the quantities which it governs. 



57] of two Algebraical Functions. 499 

The part of i2i_] within the sign of summation is 

ZiX^-i - X (Aft^, + /ie,,, + . . . + hj r (he,, J% . . . h) «"-'-' + &c., 
say ZiX'"'~^ — Zix'^~^~^ + &c., 

and the part of Ri^ within the sign of summation is 

^i_l *■»-'+! - Z'i_-yX^-i + &C., 

and 

Z-_ a;"~*+^ — Z'_ a;"~* 
Za^ ' ' ^ , — „/ ' \ , — = Zi_.ZiX + (Zi_.Z/ — ZiZ'i_i) + an algebraic fraction. 
ZiX^~^ — Zi «"~*-i ^ ' ° 

X {^i_i^i« + (Zi_,Zc' - ZiZ'i_,)} 

^Z'^,Z\_, Z\_,...Z'^i, 
Zi ~Z\_,Z\_,...Z\^^,'' 

Ti denoting Zi_,ZiX + {Zi_,Zi - ZiZ'i_,). 

Art. (e). If the process of obtaining the successive quotients and 
residues be considered, it will easily be seen that each step in the process 
imports two new coefficients into the quotients, the first quotient containing 
no literal quotient in the part multiplying x and containing the first literal 
coefficient in the other part, the second quotient containing two literal 
coefficients in the one part and three in the other, and in general the I'th 
quotient containing 2i—2 of the letters in the one part and 2i— 1 of them 
in the other. Hence Ti being made equal to Ltx + Mi; Li contains 2i — 2 
and Ali contains 2i — 1 of the literal coefficients of fx. 

Moreover, we have Zi of the form 

rr,^ Pi-2-mP i 

■Li p . 

Pi-2 = 2f (/ie,, hg^... he,_,)r}e^'r]ei+i---Vem 
and Fi, which is the ith simplified residue, vanishes when the n roots in any 
manner become reduced to only i distinct groups. 

I proceed to show that if we make 

AiX+Bi=Ui^ A\:,{x - K) + A\,(x -h,) + ...+ A\n (x - K), 
w^here in general 

Ai^ e represents X^{he^, hg^... he,_,) {h, - he[) (h, - he,) ...(K- he,_,), 

then will 

Ti= Ui. 

32—2 



500 On a Theory of the Syzygetic Relations [57 

It will be observed that Ai_e is identical with what the simplified 
denominator of the (i — l)th convergent becomes when we write he in place 
of «, and consequently, when arranged according to the powers of h^, will be 
of the form 

CiV~^ + Cjle~^' + . . . + Ci, 

where Cj, Cg... Ci are functions of the coefficients, but containing no more of 
them than enter into Qj_i, that is, containing only 2i — 2 of them. 

Now Ai is made up of terms, each consisting of some binary product of 

Ci, C. ...Ci, 

combined with some term of the series 

th'^-"; SA='■-^..2/^^ 

and any one of this latter set of terms expressed as a function of the coeffi- 
cients oifx contains at most 2i — 2 of them. 

Hence only 2i— 2 of the coefficients enter into A^, and in like manner 
only 2t - 1 of them into Bi. 

The number of letters, therefore, in Ai and in Bi is the same as in Li 
and in Mi, namely 2i — 2 and 2i— 1 respectively. 

Now let the roots consist of only i distinct groups of equal roots, so that 
Ti becomes = Zi ^^ . 

I shall show that in whatever way the equal roots are supposed to be 
gi'ouped upon this supposition, there will result the equation 

Ti=Ui, 

p._ 

■where Ti = {t^(ve,, Ve^--- %)p W— ' , 

-t i— 1 

-Pi-2 = S {»79i'79i+, ■■■Ven ^(Ve,, Ve^--- VOi-i)}, 

Pi-i = S [vei+iVei+i---V0„ ?{ve,, Vb, ■■■ VOi)}, 

and Hi = Ai'% + A^ij^ + ... + A^^'vu, 

Ae meaning t {{rje - rjg^) (rje - ve,) ■■■iVe- Ve,.i) ^ive,, Ve,--- Vet-i)}, 

and rja meaning x— k^. 

Let the n factors be constituted of m, factors %, m^ factors rj^.-.Ttii 
factors Tji. Then 

where fi=mim2...mi, 



57] of two Algebraical Functions. 501 

and ■ Pi_o = /ii^C??,, Vi---Vi) r/i™'!?.™'-' ... 77;™-i 

+ H-2^{Vi>V3--- Vi) Vi™'"'^ -^2'"- • • • Vf"'"^ 
+ &c. &c. 

+ H'iKiVi, V-2--- Vi-i) Vi"''~^V2"'-~^ ■ ■ • Vi"'', 

where n. =J— a— — ... ai = — . 

nil ^^2 ^i 

Hence 

-'» = M i('?i. '/a--- Wi ~ 1- — 1-... + - 



mi 
Again, in Ui the term containing t]^ will be 

'«i'7i {S (i?! - V2) (Vi -Vs) ■■■ (Vi - Vi) UV2,V3--- Vi)Y 
= «ii Vi X ('"2 'Ws • • • mi)"- X (771 - 772)= (tji - 7];)- ...(vi- Vi)- \HV2,V3--- Vi)}- 

= £- J/i X ^(V^, v-2 ■■■ Vi) ^{V2, Vi---Vi)- 
'"1 

Hence 



Hence, therefore, Ui — Ti vanishes whenever the roots of fx contain only i 
distinct groups of equal roots, and it has been shown that Ui and Ti each 
contain only 2i - 1 of the coefficients of fa, so that f/j — Ti is a function 
only of ir and these 2i — 1 letters, and consequently, by virtue of the Lemma 
in Art. (a), Ui — Ti is universally zero, that is, Ui is identical with Ti, as was 
to be proved. In the same manner, as observed in a preceding note 
[p. 494], the expression given in the antecedent articles for the numerator 
of the ith convergents, having been verified for the case of the roots consist- 
ing of only i distinct groups, could have been at once inferred to be generally 
true by aid of the Lemma above quoted. 

Art. (/). Since the coefficient of x in Ti is Zi_i x Zi, we deduce the 
unexpected relation 

2f (/ii, h,...hi_i) X t^ih, h ... hi) = P;- + P/ + ... + Pn\ 
where Pe = t [{h - h) {h - h^) .-{K- he,_,) K{K, he, . . . he,_,)}. 

f'x 
So that every simplified Sturmian quotient to -^-j- , when the n roots oi fa 

fx 

are real, will be the sum of n squares. But the equation is otherwise 

_ n(n-l)...{n-i+2) 
remarkable, in exhibiting the product of the sum of ^^2 ({- 1) 

squares by another sum of "^ '^~ ^'"'■""'^ ^ squares under the form of 

the sum of n squares. ; 



502 



On a Theory of the Syzygetic Relations 



[57 



If we denote the ith simplified denominator to the Sturmian convergents 

f'x 
to*^ by DiX, and if we call the t'th simplified quotient XiX, we have 



> 



XiX=^{Di_Xf{oo-h,). 



If we construct the numerators and denominators of the convergents to 
1111 



according to the general rule for continued fractions, as functions of Qi, Q^, 
kic, so that calling the denominators Aj, Aa, A3 ... A;, 



Ax=Q, A, = i 



,-l...Ai = QiAi_,-Ai_3, 



we have 



^i-lX = 



Z-i_^Z''i_i ... i^^(i_i) 



2'i-,Z\ 



z\.. 



A- 



Ai_i« being in fact the multiplier oi f'x in the equation which connects fx 
and/'a; with the (i — l)th complete residue, and consequently, retaining Q(«) 
to designate the complete I'th quotient, we have 



Qi{x) = 



Z\_^ Z\_^Z\. 



Zi Z\_,Z\ 



..Z' 



Z\_^Z\^,Z\_^...Z\ 
Zi Z\^Z\_....Z\ 



t[Dt-AY{«>-h) 



-^[^i^,heY{x-K). 



which equation gives the connexion between the form of any quotient and 
that of the immediately preceding convergent denominator of the continued 

fraction which expresses"^ . 
fx 

Art. {g). I have found that the coefficients of the n factors of fx in the 
expression above given for the quotients possess the property that the sum 
of their square roots taken with the proper signs is zero for each quotient 
except the first (the coefficients for the first being all units), that is 
JDiJii + Dih3 + ... Dihn = for all values of i except i=l. Moreover I find 
that the determinant formed by the n sets of the n coefficients of the factors 
of fx in the complete set of ?i quotients is identically zero, that is, the 
determinant represented by the square matrix 



= 0. 



1, 


1, 


1, 


1, 


(DAT, 


(DAY, 


(DAY 


...(DAY 


(DAT, 


(DAT. 


{DAY 


...(DAY 



(D^-AY, (Dn-AY, (Dr,-AY-iDn-AY 



57] of two Algebraical Functions. 503 

Art. (/i). It should be observed that Ui is the form of the simplified 
quotients for all the quotients except the nth. (that is, the last), for which 
the simplified form is not Un, but Un-^^{hi, h^ ... hn), which arises from the 
circumstance of the last divisor, which is the final Sturmian residue, not 
containing x; it being evidently the case that the division of a rational 
function of a; by another one degree lower, introduces into the integral part 
of the quotient the square of the leading coefficient of the divisor, subject 
to the exception that when the divisor is of the degree zero, the simple power 
enters in lieu of the square. The general formula gives for the reduced 7ith 
quotient the expression 

2 [Qh - k), (h, - h) . . . Qh - hn) ^(h„ h,... h^)Y {x - M 

which equals 

^(Kh... hn) t^(k,, h... hn) {X - h). 

Rejecting the first factor, we have 

S^(/l2, A3... A„) («-Ai), 

which is equal to the penultimate residue, which residue is (as it evidently 
ought to be) identical with the simplified last quotient. 

Art. {i). We have thus succeeded in giving a perfect representation 

of "^ , that is, of 
fa 

11 1 

X — hj X — h^ X — hn 

under the form of a continued fraction of the form 

11 1 



mi (x — 61) — nii, (a; — 62) — nin {x — e„) ' 

where m^, m2 ... mn', e^, e^.-.en ai-e all determinate and known functions 
of /ii, /12 ... hn. 

We may by means of this identity, differentiatibg any number of times 
with respect to x both sides of the equation, obtain analogous expressions for 
the series 

1 1 1 

(x - hy '^{x- KY + •■•+(«- hnj • 

But to do this we must be in possession of a rule for the differentiation of 
continued fractions whose quotients are linear functions of the variable. 
I subjoin here the first step only toward such investigation. 

Let the denominator of 

11 1 



504 On a Theory of the Syzygetic Relations [57 

where q^, q^-.-qn are anyw arbitrary quantities, be denoted by [q^, q.,, qs-..qn], 
so that the entire fraction will be equal to 

[^2, g-s... qn] 



1i>q2,qz---qn]' 



Any such quantity as [qi, qt^^ ■■■ qn] niay be termed a Cumulant, of which 
qi, qi+i ...qn may be severally termed the elements or Components, and the 
complete arrangement of the elements may be termed the Type. The 
cumulant corresponding to any type remains unaffected by the order of the 
elements in the type being reversed, as is evident from any cumulant 
being in fact representable under the form of a symmetrical determinant, 
thus, for example, the cumulant [^'i, q.2, q-^, ^j] may be represented by the 
determinant 



?1. 


1, 


0, 





1, 


q2, 


1, 





0, 


1, 


?3, 


1 


0, 

ma 


0, 

oner 


1, 

be r 


?4 

spre 


94, 


1, 


0, 





1, 


23. 


1, 





0, 


1, 


^2, 


1 


0, 


0, 


1, 


qi 



and [^4, qs, q^, q^] will in like manner be represented by the determinant 



which is equal to the former. 

Art. (j). Let it be proposed in general to find the first differential 
coefficient in respect to x of the fraction 

[qi,q„q,...qn\ 
where each g is a function of one or more variables. 

I find that the variation of Fi may be expressed as follows : 
- BFi = {S [q„ q„_... qi_„, q„] + S[q^,q^... qi_^, q^-i] qn^ 
+ S [qi, q2, gs . . . gi-a, qn-2] [qn, qn-if +■■■ 
+ S [qu q2, ?3 •■• g£-2> qi-i] [qn, qn-i, qn^..- qif} 



57] of two Algebraical Functions. 505 

Art. (/<;). Suppose i=2, and gi=ai« + 6i, 52 = «2« +-&2 •■• 2'n = ««* + &», 
we shall have by virtue of the above equation, 

d ^ ^, ^ . d { \ 11 1 

T- F^, that IS ^- \ . . . — 

ax ' ax [qi — qo— qs qn 

= "fvT^;^ W~v i"»^' + "'^-^In + ^^"-2 [g,., 2«-i]' + &c. 

|_(/i , q2 ■■■ qti] 

+ a,i[qn, qn-i, g„-o...g,p}. 

If we call ¥2=%- every such quantity as [g,,,, 5'„_i...5'j] represents to a 
jx 

constant factor pres the {i — l)th simplified residue {<^x counting as the first 

of them) to -V > and making certain obvious but somewhat tedious reductions, 
' fx' "= 

and rejecting the common factor — . , we obtain the expression 

CojRi" il.," -Rq^ -R71" , /., ,ijy 

—rr- + TTfr + -cTrr + ■ • • + 7< — 7^ = ^«/ « - ^A. 

where jRj, i?2 ••• -R™ represent (/>« and the successive simplified residues 
toy«, ^x, while Gi means tlie coefficient of the highest power of x in i?j, and 
C„ the first coefficient va.fx*. 

Art. (Z). If we take gx of the same degree as fx, and for greater 
simplicity make the first coefficients in fx and gx, each of them unity, 

* This result may be obtained directly as follows : — 

Let /x, (px and the (m-1) complete Sturmian residues be called p^, Pi, po ■••?»; 1^* t^* ^ 
complete quotients be called q^, q^ ... g„ , and let the allotrious factors to the residues p.,, p^... p^ 
be called /j^, /i^ ... p^; then 

Po=<liPi-P2' Pi = Q-2Pi-Ps< P2 = 1sPs-Pi> &<=.; 
hence pjp„- pf,Sp^ = p^bq^ + (p.^Sp:^- p^Sp.^ 

=Pihi + PiH2 + {Ps^Pi - PiSps) 

= &o. 

= pi" Sqi + p.- Sq„ + Pa" 5?3 + . . . + p„2 Sq^ ■ 
but we have in general pi=fijEi, 

hence 5q(= -^ ^^ dx, 

C ■ 
and p^Sq^=—^ p.^-■^^p.^ItfSx•, 

but it may be easily seen that 

ft-iMi=7rt — > except when i = l, for which case /t;_i/i; = l, 

hence Pi^i = n — Tr-R/^a;, when i>l, and =-:^ Rj^Sx when i = l, 

which proves the theorem in the text. 



506 On a Theory of the Syzygetic Relations [57 

the successive simplified residues to y- will be identical with the simplified 

residues to -^ — ^^^— (including amongst them the quantity gx —fx itself), 
gx 

and, since 

{fx - gx} g'x - {fx - gx]' gx = g'xfx -f'xgx, 

the right-hand side of the equation above written, when the residues, instead' 
of referring to / and <^, are made to refer to / and g, taken of the same 
degree in x, becomes equal to f'xgx — fxg'x ; and if we now agree to 
consider/ and g as homogeneous functions each of the ?ith degree in x and 1, 
the equation becomes 

ni Rl^ Ri_ R^ 

Gi C1C2 C2C3 On-iC,i 

= 9(^,l)^J(a^,l)-fia=,l)^g(x,l) 

dx^ dl^J\dx-'J n\ dx-' dl-' J\dx^J 

<Vd9_¥dg\_\T(f„. 
dxdl dldx\ n^-''^'' 

where J indicates the Jacobian of the given functions / and g in respect to 
the variables x and 1, meaning thereby the so-called Functional Determinant 
of Jacobi to f and g in respect of x and 1, which equation also obviously 
must continue to hold good when we restore to the coefficients of «" vnf and 
g their general values. 

It may happen that for particular relations between the coefficients of 
f and g certain of the residues may be wanting, which will be the case 
when any of the secondary Bezoutics have their first or successive terms 
affected with the coefficient zero; the equation connecting the residues 
with the Jacobian will then change its form (as some of the quantities 
Oi, C2 ... On will become zero) ; but I do not propose to enter for the present 
into the theory of these failing, or as they may more properly be termed, 
Singular cases in the theory of elimination. 

Art. (m). The series last obtained for J (f, g) leads to a result of much 
interest in the theory, and of which great use is made in the concluding 
section of this memoir, namely the identification of the Jacobian (abstraction 
made of the numerical factor n) with what the Bezoutiant becomes when in 
place of the 71 variables in it, Mj, u^ ... u^, we write a;"~^, a;"~^ ...x, 1. Thus 
suppose /and g to be each of the third degree, and let 

Ax' + Hx+ G, 

Hx"- + Bx+F, 

Ox' + Fx + C, 



57] of two Algebraical Functions. 507 

be the three primary Bezoutics ; if we make 

x^^u, X — V, 1 = 10, 

these may be written under the form 

Au + Hv-\- Gw = L, 

Hu + JBv+Fw = M, 

Gu + Fv +Cw = N, 

and if the Bezoutiant be called 3, we have 

^^dS ^^dS^ ^^dS_ 
du' dv ' dw 

The simplified residues to / and g are L, (L, M), (L, M, N), where (X, M) 
means the result of eliminating u between L and M, and {L, M, If) the result 
of eliminating u and v between L, M, N\ and by a theorem (virtually implied 
in the direct method* of reducing a quadratic function to the form of a sum 
of squares), if we call the leading coefficients of these quantities Oj, G^, C^, 
we have 

X^ jL, Mf (L, M, N) ^ 

ri "r r< p "r r< rt "• 

Hence, when ?i = 3, lJ{f, g) = S when in 9", u, v, w are turned into a;^ x, 1 ; 
and so in general for any values of n, the Bezoutiant correspondingly modified, 

becomes - J {f, g), as was to be shown -f. 

f'x 
Art. (n). The expressions obtained for the quotients to -^r- may be 

generalized and extended to the quotients to ^ , where (j)X and fx are two 

functions of x of any degrees m and n, whose roots are respectively, ki,hz...hm, 
and Ai, /(-a . . ■ hn- If we suppose 

6x \ 1 1 1 



/a; Q(«)- gaW-^s («)-'" g'm+iC*)' 

where Q{x) is of ?i — m dimensions, and ga («). 23 («) ••• 2'm+i(«) each of one 
dimension in x, it may be proved that on writing 

1 1 1 ^ Nj {x) 

Q(x)- q, {x) -''" qt (x) Di (x) ' 

* Namely, that of M. Cauchy, adverted to in Section IV. Arts. 44—45. [p. 511 below.] 

t Compare Jaoobi, De Eliminatione, § 2. The general expression for the allotrious factor, 

I may here incidentally mention, is given under the head Theorem a, § 16, which comes quite at 

the end of the same paper. 



508 On a Theory of the Syzygetic Relations [57 

we shall have 

^. fh 



i^ [{Dihefp^^ (*■ - h^ = G'9i+i {«^\ (B) 

where C + C = 0, (E) 

Cqi+i{x) being the (i+l)th simplified quotient. When Q (a;) is a linear 
function of x, in finding q-^^x from the formula (B), we must take D^ = 1. The 
proof of this theorem being generally true, may easily be shown to depend 
upon its being true in the special case*, when m = fi + i, and ?i = /x + i' 
(m being supposed less than n), and Aj, h^... h^ become l^, l^... l^, h^, h^... hi; 
while k^, k^-.-kja become Zj, l^ ... i^, k\, k^... ki; and the truth of the theorem 
for this special case (if for instance we wish to prove the formula (B)) depends 
upon the expression 

/Jij, h-.. Aj'-A ^ fK , h ■■■ h'-A 
Ui, h...k,n J ' Ui', hi'+i...h„ 

hj, As ••• hi'\ /hi, h^ ... hi' 

ki, ks...kmJ ' W+i, Ai'+2...A; 

being identical with the expression 

t:t:t^(t:t.:::t-^<'''- '■■)<*'-''='•••*-'-' 
*, \ 



VAi, Aa ... A;._i Ai'+i ... A„ 

as it may readily be shown to be. And the formula (A) may be verified 
in precisely the same manner. There is no difficulty in finding the values 
of C and C, which are products of powers, some positive and some negative, 
of the leading coefficients in the simplified residues, and recognising that 
they satisfy the equation (E) ; when ^x is of one degree below /« this equation 
is of the form (7 + 0' = 0. 

Art. (o). When ^x=f'x, this expression for the (t + l)th simplified 
quotient becomes '2<{DiKf{x — h), as previously found; the correlative ex- 
pression will be 

-%{Niky^^ix-k), 

* By yirtue of the Lemma, that when <px and fx are two algebraical functions, no function 
of the coefficients vanishing identically when i roots of /r coincide with i roots of <px respectively 
can be formed, in which there are fewer of the coefficients of / and respectively than appear 

in the leading coefficient of the (n-i + l)th residue of -. . 



57] of two Algebraical Functions. 509 

k being any root of f'x = 0, which is equal to the former expression. The 
general expressions above given for the simplified quantities are of course 
integral functions of h and k, although given under the form of the sums 

of fractions, by virtue of the well-known theorem that 2 jjr , where ^ is an 

integral function of li, and the summation comprises all the roots (h) of 
fh = 0, is always integral. 

Art. (p). It will be found that for all values of i greater than unity 

e=i (p i^e 

and that I (Dihg) ^ = 0. - 

The theorem of Art. (n) is in effect a theorem of cumulants of the form 

[ft («)> qo. («)... qi (*•) ...qn («)], 

where the elements are all independent of one another, and 

fx = [ft («), q^ («), ^3 (x) ... q^ (x)], <f)X = [q^ (x), q^ {x) ... q^ («)], 

n being any number whatever greater than i; this makes the theorem still 
more remarkable. The urgency of the press precludes my investigating 
for the present the more general theorem which must be presumed to exist, 
whereby qi+i can be connected with [^i, q^, q^ ... qi], or [g.^, q^... qi\, and with 
[ffi. 2'2, 9s ••• qi+e\ and [q^, q^... 9;+ J, when each q represents a function of an 
arbitrary degree in x. The theorem so generalized would comprehend the 
complete theory of the quotients arising from the process of continued 
division, without exclusion of the singular cases (at present supposed to be 
excluded) where one or several consecutive principal coefficients in one or 
more of the residues, vanish. 

Art. {q). The complete statement of two twin theorems suggested by 
and intimately connected with the biform representation of the quotients 

~- , given in the preceding article, is too remarkable to be omitted. 

f'x 
Suppose ^x =f'x, and let the successive convergents to "^ be called 

/« 

T\x' T^'"T^' ~T^' 

where the subscript index to ^ or T indicates the degree in x. Then if we 
call the roots oi fx,\,lu ...hn, the theorem already cited in a preceding 



510 



On a Theory of the Syzygetic Relations 



[57 



article, concerning the denominators of the convergents, may be expressed 
as follows : — 

(f%y (f%y (f%x- 

\4>hJ ' \4>hJ ■•■ \4>hJ 
(Zky, {Tji,y ...{T,Ky 
(T,hy, (Tjuy ... {T.hnf 



{Tn-Xr, {Tn-M-{Tn-,Kr 



= 0, 



where it will be observed that the first line of terms consists exclusively 
of units, since f'x = ^x by hypothesis. 

Correlatively I have ascertained that preserving the same assumption 

that j)X=f'x, so that consequently ^ means "^27- > the following theorem 

/"^ Jk 





m 




%(h)Y, 


um-- 


..[t.{kn-.)Y 


{t.(h)Y> 


[k{h)Y 


...{t,{kn-.^)Y 



= 0. 



{tn-.{h)Y, [tn-.{k.:)Y:.[tn-.{kn-^)Y 

It may consequently be conjectured, when (/> and f are independent functions 
of X and respectively of the degree n — 1 and n, and ^ is expanded under 

the form of a continued fraction, of which, as before, ^, ^...-2f? are the 

successive convergents, that we shall have analogous determinants to the 
twin forms above given, each separately vanishing, these more general 
determinants differing only from their model forms in respect of the upper- 
most line of terms in the one of them, being each multiplied by certain 
functions of h-y, h^ ... hn respectively (all of which become units when (^x =_/'«), 
and in the other of them by certain functions of k^, k^... kn. 

The exact form, however, of such functions, and even the possibility 
of such form being found capable of making the determinants vanish, remains 
open for further inquiry. 



57] of two Algebraical Functions. 511 

Section IV. 

On some further Formulce connected with M. Sturm's theorem, and on the 
Theory of Intercalations, whereof that theorem may be treated as a 
corollary. 

Art. 44. As preparatory to some remarks about to be made on the formuloe 
connected with M. Sturm's theorem, it is necessary to premise two theorems 
of great importance concerning quadratic functions, one of which, notwith- 
standing its extreme simplicity, is as far as I know very little (if at all) 
known, and the other was given in part many years ago by M. Cauchy, but 
is also not generally known. The former of these two theorems is as follows. 
If a quadratic homogeneous function of any number of variables be (as it may 
be in an infinite variety of ways) transformed into a function of a new set of 
variables, linearly connected by real coefficients with the original set, in such 
a way that only positive and negative squares of the new variables appear in 
the transformed expression, the number of such positive and negative squares 
respectively will be constant for a given function whatever be the linear 
transformations employed. This evidently amounts to the proposition, that 
if we have 2n positive and negative squares of homogeneous real linear 
functions pf n variables identically equal to zero, the nvimber of positive 
squares and of negative squares must be equal to one another, so that 
for example we cannot have 

Ui' + «/+... + M«^ + il\+i - u\+^ — u\+3 - ... - m\„ 
identically zero when n of the variables are linear functions of the remaining 
n; and this is obviously the case, for if the equation could be identically 
satisfied we might make 

'^n+2 ^^ ^ij ^n+3 ^^ '^^2 ■ • • '^^2n ^^ '^n—ii 

and we should then be able to find u^+i as a real numerical multiple of m„, 
and consequently should have the equation u.^ {1 + A-j = 0, which is obviously 
impossible; a fortiori we may prove that in the identical equation existing 
between the sum of an even number of positive and of negative squares 
of real linear functions of half the number of independent variables, there 
cannot be more than a difference of two (as we have proved that there cannot 
be that difference) between the number of positive and negative squares. 
Hence there must be as many of one as of the other ; and as a consequence, 
the number of positive squares or of negative squares in the transform of a 
given quadratic function of any number of variables effected by any set of 
real linear substitutions is constant, being in fact some unknown transcen- 
dental function of the coefficients of the given function. I quote this law 
(which I have enunciated before, but of which I for the first time publish 
the proof) under the name of the law of inertia for quadratic forms. 



512 On a Theory of the Sijzygetie Relations [57 

Art. 45. The other theorem is the following. If any quadratic function 
be represented in the umbral notation* under the form of 

(a-^X-y + a«X2 + . . . + UnXn)", 

where Oi, a^.-.a^ are the umbrse of the coefficients, and x^, x^...Xn the 
variables, then by writing 

I tti I \ 0'%\ I I3 I I ^^4 I I "■« I 

«2 + M^3+ ^\Xi+...+ \xn = yi, 

I a„ as I I «!, tts I I 0,1, O4 1 I "^. <^n I 

I a,, a^, as \ I a,, a,, a, I I a^, a^, a^ I 

I Oi, 02,03! \ai,a.2,ai\ |ai,c(2, an| 



&c. &c. &c. 



tti, a^ 

\<h, 0,2 



^n — yn> 



(ai«i + a2«2 + ••• + ««««)" will assume the form 

^1 ) ^2 ^1 > ^2 J ^3 ^1 ) ^2 * • • ^tl — 1 } ^n 

1 Oi I ttj ai, tta ffli, aa ... a„_i 

1 ffli I I ai , a2 I \ tti, a2 ... o,n-i \ 

and consequently the number of positive squares in the reduced form of the 
given function will always be the number of continuations or permanencies 
of sign of the series 

I tti I 1(^1, do I . I Cti, a2, «3 I \ Oi, 0.2 ■■• Ctn\ 

' I Oil' !»!, 02!' I Oi, tta, as I '" I Oi, tta ... a„ I' 
the several terms of this progression being in fact the determinants of what 
the given function becomes when we obliterate successively all the variables 
but one, then all but that and another, then all but these two and a third, 
until finally, the last term is the determinant of the given function with 
all the variables retained. This comes to saying that if we call the function 
(suppose of four variables)/, and write down the matrix 
d'f dY df d'f 



dx^' 


dxjdx^' 


dx^dxs ' 


dx-^dxi' 


d^ 
dx^dxi' 


df 
dx^^' 


df 
dx^dxs ' 


df 
dx^dxi ' 


d^ 
dx^dx-i ' 


df 
dx^dx^ ' 


df 
' dx,'' ' 


dlf 
dx^dxi ' 


d^f 
dxidXi ' 


df 
dxidx^ ' 


df 
dx^dxs 


df 
' dx^^ ' 



* For an explanation of the umbral notation, see London and Edinburgh Philosophical 
Magazine, April 1851, or thereabouts [p. 243 above]. 



57] 



of two Algebraical Functions. 



513 



(where all the terms are of course coefficients of the given function expressed 
as above for greater symmetry of notation), the inertia of/ will be measured 
by the number of continuations of sign in the series formed of the successive 

principal minor coaxal determinants (in writing which I shall use in general 



(r, s) to denote 



dXfdx. 



,)■ 



(1. 1). 



(1, 1), 
(2, 1), 



(1.2) 
(2,2) 



(1. 1). 


(1, 2), 


(1.3) 


(2, 1), 


(2, 2), 


(2, 3) 


(3, 1), 


(3, 2), 


(3. 3) 



(1, 1), (1, 2), (1, 3), (1, 4) 

(2, 1), (2, 2), (2, 3), (2, 4) 

(3, 1), (3, 2), (3, 3), (3, 4) 

(4, 1), (4, 2), (4, 3), (4, 4) 
and in like manner in general*. 

Art. 4G. Reverting now to the simplified Sturmian residues, since by 
the theory set out in the first Section these differ from the unsimplified 
complete residues required by the Sturmian method only in the circumstance 
of their being divested of factors which are necessarily perfect squares and 
therefore esseutially positive, these simplified Sturmians may of course be 
substituted for the complete Sturmians for the purposes of M. Sturm's 
theorem. The leading coefficients in these simplified Sturmians, reckoning 
/' (x) as one of them, will be 

mt^(hi, hi), "Z^ihi, h, hs) ... f (/ii, h^... km), 
which it is easily seen, as remarked long ago by Mr Cayley, are the successive 
principal minor coaxal determinants of the matrix 

0"o, O"], ""a. 0-3 ■■• 0"7n-i, 



""m— 1) ""m •■■""am— 2. 

* I have given a direct a posteriori demonstration in the London and Edinburgh Philosophical 
Magazine, that the number of continuations of sign in any series formed like the above from a 
symmetrical matrix, is unaffected by any permutations of the hnes and columns thereof, which 
leaves the symmetry subsisting, that is to say (using the umbral notation), if 9-^, 6^, 6^ ... ff^ are 
disjunctively equal, each to each, in any arbitrary order to 1, 2,S...i, the number of continua- 
tions of sign in the series 



1, 



% I ' I "91, "92 r I Hi> "92. "■esl'"' I "91. "92. "93 ••■ "'ft I ' 

is irrespective of the order of the natural numbers 1, 2, 3 ... i in the arrangement $1, 8^, 6^... 6f, 
s. 33 



514 On a Theory of the Syzygetic Relations [57 

where in general o-, = Ai*" + Aa*" + . ■ • + h^, and of course o-q = m. M. Hermite 
has improved upon this remark by observing, vfhat is immediately obvious, 
that if we use o-^ to denote, not the quantity above written, but 

the successive coaxal determinants of the above matrix will become re- 
spectively 

^ 1 vf ^{K,h) 



X — Jh' \{^ - h) (« — ih)) 

(x — hj)(x — As) {x — h^''"{x — Ai) {x — h^ ...{x— hm) ' 

that is to say, these successive coaxal determinants, when multipKed up by 
fx, will become respectively 

S (a; - A„) {x-h,)...(x- AJ, 2^(Ai, h^) [{x-h,) (x-h,) ... (x-h^)}, ... 

Sr(Ai,A,...A™), 

that is to say, will represent the simplified Sturmian series given by my 
general formulae. M. Hermite further remarks, that the matrix formed 
after this rule will evidently be that which represents the determinant of 
the quadratic function (which may be treated as a generating function) 

S r {Mi + A1M2 + Ai^Ws + . . . + Ih™~'^u,n}-, 

a; — Ai ' 

in which, since only the squared differences of the terms in the (A) series 
finally remain in the successive coaxal determinants, we may write (x — hi), 
(«— A2) ... (x — hm) simultaneously in place of Aj, Aj ... A^ without affecting the 
result ; consequently the generating function above may be replaced by the 
generating function 

2 —^j- [ui + (a; - Ai) «2 + {x- h^)- U3 + ... + {x- Aj)'""! «m}^ 
the corresponding matrix to which becomes 

x — hi 



57] 



of two Algebraical Functions. 



515 



1 f'x 

where 6i denotes S (a; — hy-, and 2 j- =-^ . Hence every simplified 

residue is of the form 



f'xy. 



+fxx 



0, e„ e,... Or 



Or, Or 



The residue in question will be of the degree m — r — 2 in x, and consequently 
we have, according to the notation antecedently used for the syzygetic 
equations 

^1, ^2 ■■■6r 



^r t "r+l ■ • • "'ir—l 

0, 00, e. 



d,., dr+i 

Elegant and valuable for certain purposes as are these formuliB for t^+i 
and Tr, they are affected with the disadvantage of being expressed by means 
of formulae of a much higher degree in the variable x than really appertains 
to them, the paradox (if it may be termed such) being explained by the 
circumstance of the coefficients of all the powers of a; above the right degree 
being made up of terms which mutually destroy one another; upon the 
face of the formulae, tr+i and t^ which are in fact only of the degrees r + 1 
and r respectively in x would appear to be of the degree 



l + 3+o + ... + (2r-l), 



that is of the degree r^ 



Art. 47. I may add the important remark, which does not appear to 
have occurred immediately to my friend M. Hermite when he communicated 
to me the above most interesting results, that in fact, by virtue of the law 
of inertia for quadratic forms, we may dispense with any identification of the 
successive coaxal determinants of the matrix to the generating function 

2 r [ui + hiU^ + h^Ui + . . . + h^-^UmY 

with my formulae for the Sturmian functions, and prove ab initio in the 
most simple manner, that the successive ascending coaxal determinants 

\ 33—2 



516 On a Theory of the Syzygetic Relations^ [57 

(always of course supposed to be taken about the axis of symmetry) of the 
matrix to the form above written, or to the more general form (which I shall 
quote as (G), namely) 

2 (p - h,)i {4>, (h) u, + <!>, (AO «,+ ...+ <^„. (S) «^}^ (G) 

(where </>,, <^3 ... ^^ are absolutely arbitrary integral forms of function with 
real coefficients), will form a rhizoristic series in regard to fx (that is a series, 
the difference between the number of the continuations of sign between 
the successive terms of which corresponding to two different values of p will 
determine the number of real roots of x lying between such two assumed 
values), provided only that q be an odd positive or negative integer. Nothing 
can be easier than the demonstration, for whenever p is greater than any 
one of the real roots as Aj : — 

Firstly, any pair of imaginary roots will give rise to two terms of the 
form 

(Z + TO V- 1)* {v-^wsl-\f and {I - in ^- 1)« (v - w^/-\f, 
or more simply 

(Z + il/ V- 1) («' - w- + 2ot<; V- 1) 
and {L-M^J-\){v^-w"--'2,vw^/-l), 

where v and w are real linear functions of Mj, u^ ... Um- The sum of which 

couple will be 

2 
2 [L (V- - w'') - 2Mwv} = Y {{Lv - Mw)- - {D + M-) w^) =p- - q"; 

so that each such couple combined will for every value of x give rise to one 
positive and one negative square. 

Secondly, any real root of the series h^, h^... h^, when p is taken greater 
than such root, will give rise to a positive square of a real linear function 

of Ml, M., ... Um. 

Thirdly, any real root of the same series, when p is beneath it in value 
(q being odd), will give rise to the negative of the square of a real linear 
function of the same. Hence the number of real roots between p taken 
equal to one value (a), and p taken equal to any other value (b), will be 
denoted by the loss of an equal number of positive squares- in the reduced 
form of the expression (G) when p is taken a and when p is taken b; 
that is by virtue of Art. 4-5 will be denoted by the difference of the number 
of permanencies of sign in the successive minor determinants of the matrix 
corresponding to the quadratic form (G)* (which we have taken as our 

* The inertia of the quadratic form (G) is the measure of the number of real roots of fx 
comprised between oo and p, and may be estimated in any manner that may be found most 
convenient. If p be made infinity, and (p^h be taken equal to hi~'^, and the inertia of the corre- 
sponding value of (G) be estimated by means of the formulie in ordinary use by geometers for 



57] of two Algebraical Functions. 517 

generating function) resulting from the substitution respectively of a and 
b in place of p, which gives a theorem equivalent to that of M. Sturm, 
transformed by my formulfe, when we choose to adopt the particular 
suppositions 

q = -\, 0i/i = 1 , 4>Ji = h, 4>,h = h\ ... c})„^h = h'^-K 

This method of constructing a rhizoristic series to fx by a direct process 
is deserving of particular attention, because it does not involve the use of the 
notion of continuous variation, upon which all preceding proofs of Sturm's 
theorem proceed. It completes the cycle of the Sturmian ideas. Happily 
this cycle was commenced from the other end, for it would have been difficult 
to have suspected that the root-expressions for the terms in the rhizoristic 
series could be identified with the residues, had the former been the first 
to be discovered, and much of the theory of algebraical common measure 
laid open by means of this identification would probably have remained 
unknown. 

Art. 48. I proceed now to consider a theorem concerning the relative 
positions of the real roots of two independent algebraical functions as 
indicated by the succession of signs presented by their Bezoutian secondaries ; 
this more general theory of intercalations or relative interpositions will be 
seen to include within it as a corollary the justly celebrated theorem of 
M. Sturm. 

Let the real roots of fx taken in descending order of magnitude be 
/ii, h^...hp, and the real roots of 4)X taken in the like order %, Tj^-.-riq, 
so that 

fx = {x — hi) (x — ho) ...{x — hp) H, 

^x = {x--ni){x-7}„^...{x-r]q)K, 

H and K being functions of x incapable of changing their signs. Now, as in 
M. Sturm's method, let us inquire what takes place in respect to the sign of 

V^, which I shall call the Indicatrix, as x descends the scale of real 

magnitude from + oo to - oo . If between + oo and /ij, i real roots of j>x are 
contained, it is obvious that as x travels from + ao to the superior brink 
of Ai, the Indicatrix will change its sign from + to - and from - to + alto- 
gether i times, so that at the moment when x is about to pass through Aj, it 

determining the nature of a surface of tlie second degree, the criteria of the number of real roots 
in fx will he, or may be made to he, symmetrical in respect to the two ends of the expression fx. 
This system of criteria, however, is not so good as that given by the Bezoutiant to the two 
differential coefficients of / (x, 1) taken with regard to x and 1 respectively, which will also 
possess the like character of symmetrical indiffereace, and be one less in number than the 
former. 



518 On a Theory of the Syzygetic Relations [57 

will be positive if i is zero or even, and negative if i is odd ; but the moment 
after x has passed through the value Aj, the indicatrix will be negative 
on the first supposition, and positive on the other supposition. Hence 
immediately after the passage of x through h^ the indicatrix will have been 
once oftener negative than positive on the one supposition, and as often 
negative as positive on the other. Again, in like manner as x traverses 
the interval between \ and the inferior brink of A.^, if no tj or an even 
number of 7;'s occupy this interval, the sign which the indicatrix had at the 
beginning of this interval will have been reversed once oftener than restored; 
but if there be an odd number of 7)'s so interposed, the number of reversals 
and restorations will have been identical ; and so for each successive interval, 
reckoned from a value for x immediately subsequent to one real root oi fx, 
down to a value immediately subsequent to the next less real root of the 
same ; and it is evident that the effect upon the sign of the indicatrix at 
the end of every such interval depends, not upon the number of 17's grouped 
together in such interval, but upon the form of the group as regards its 
being made up of an odd or even number of terms, the first interval being 
of course understood to extend from + 00 to a value immediately inferior 
to hi, and the last from a value immediately inferior to hp to — 00 . Hence 
as regards the relation of the sign of the indicatrix at the beginning to the 
sign at the end of every such interval, nothing will be altered by taking 
away any even number of tj's that may be found therein. If we suppose 
this to be done, we shall then have in some of the intervals one 7] occurring 
and in the other intervals no r] ; that is to say, some of the lis will be 
separated by single 7;'s, but other h's will come together. Again, by removing 
any even number of h's not separated by tj's (and thus removing an even 
number of intervals), it is clear that as many changes of sign of the indicatrix 
will have been done away with from + to — as from — to +, and no effect 
upon the excess of the one kind of changes of sign over the other kind of 
changes of sign will have been produced. By removing pairs of h's in this 
manner, it may happen that t^'s will again be brought together, any even 
number of which, not separated by h's, may again be removed and then pairs 
of h's not separated by ?;'s in their turn, and so continually toties quoties until 
at length we must arrive at a reduced system of h's and tj's, where no two 
h's and no two t/'s come together, or else all the h's and all the tj's will have 
disappeared. Let the scale of h's and 17's thus simplified and reduced be 
called the effective scale of intercalations. The number of h's and the number 
of rj's in any such scale will be equal, or will at most differ from one another 
by a unit, since at each part of the scale, except at the end, every h is 
followed by an tj and every t) by an h. If the scale begins and ends with an 
h, there will of course be one more h than rj ; if it begin and end with an rj, 
there will be one more 77 than A ; if it begin with an A or an 77 and end with 
an Tj or A, there will be as many of the one as of the other. 



57] of two Algebraical Functions. 519 

Firstly, suppose the effective intercalation scale to commence with an h ; 
then in passing from + co to just beyond the first h the sign of the indicatrix 

^ changes from + to — ; it changes again from — to + as it passes the first 

r), then again from + to — as it passes the second /i, and so on ; that is to say, 
there will be a change always in the same direction from + to — as a; passes 
from being just greater than to being just less than any h appearing in the 
effective scale. Secondly, if the effective scale begin with 7?, the indicatrix 
will conversely be negative after passing the first and every subsequent 
77, and change from - to + in the act of passing through the first and every 
subsequent h. So that on either supposition the changes of sign for the 
effective scale always take place in the same direction, and the number 
of A's in the effective scale will be measured by the number of such changes, 
and consequently will be measured by the difference between the number 

of times that the indicatrix 2_ changes its sign from + to — as a; passes 

through each in turn of the real roots of /*•, and the number of times that 
in passing through any such root it changes its sign from — to + ; if the 
former number be greater than the latter, the effective scale of interpositions 
will begin with a root of fx ; if it be less, the scale will begin with a root 
of (f}X. If instead of beginning with + 00 and ending with — 00 we begin and 
end with any two limits, a and h respectively (making abstraction of all roots 
of fx or of </)« lying outside these limits, and forming the effective inter- 
calation scale with the roots comprised within these limits exclusively), 
we shall obviously obtain a similar result, but with the condition that the 
changes from + to — will be in excess if an even number of /I's and tj's 
combined be cut off by the superior limit, and the effective scale begin with 
an h, or if an odd number of lis and t^'s combined be so cut off and the scale 
begin with an 7; ; and in defect if an odd number of h's and 77's combined 
be so cut off and the scale begin with an h, or an even number be so cut off 
and the scale begin with an rj. If, now, supposing /*• to be of ??, and 4>x 
of not more than n, say m dimensions, we form the signaletic series fx, j>x, 
J3i, Bi... Bm (where the B^,B^...Bj„ are the Bezoutian secondaries or simplified 

successive residues corresponding to ^ expanded under the form of an 

improper continued fraction), it may be shown, in the same way as for 

Sturm's theorem, that whenever ^ changes from + to - a change of sign 

will be gained in the series, and when from — to + a change will be lost ; 
and that no change can be gained or lost except as x passes through the 
successive real roots of fx. Hence the difference between the number of 
changes of sign in the above signaletic series when x is taken a, and the 
number of the same when x is taken h, will indicate the number of roots 



520 On a Theory of the Syzygetic Relations [57 

of fx remaining in the effective scale of interpositions formed between such 
of the roots of fx and of jix as lie between a and h ; calling the one 
number / (a) and the other / (6), the sign of I (b) — I (a) depends not on the 
relative magnitudes of a and b, but upon the manner in which the 
effective scale commences; ii 1(a)— 1(b) is positive, the effective scale 
formed between the a and b will commence with a root of fx ; if negative, 
it will commence with a root of <jix. 

Art. 49. In forming the scale of effective interpositions, it is evidently 
not necessary to go on reducing the h series and the t? series separately 
and alternately ; the same result will be effected more expeditiously by 
eliding simultaneously any even number of h's that come together without 
being separated by an 77, and any even number of t^'s that come together 
without being separated by an h, and, repeating this process of simultaneous 
elision, as often as may be required, until no two h's or t^'s come together. 
Thus, for instance, denoting the magnitudes of the series of real roots of 
/and of (j) by the distances of /;. and tj points taken along a right line from 
a fixed point therein, and supposing such series of roots between the limits 
a and b to be 

khhr] 7] Tj JiTj 7} InjTj t] h h 7) h r) h hh h h 7) 7] h, 

our first reduction brings this scale to the form 

IiT) h hrj 7] h7] h h; 

the next reduction brings it to the form 

h7] r)7]h7) ; 

and a third and final reduction brings it to the form 

h7]h7j ; 

and accordingly we shall find for such an arrangement of the h and r] 
system 

I(b)-I(a)=±2. 

dfx 
Art. 50. If we suppose (jjx = -=- , by a well-known theorem of algebra,, 

any two consecutive roots of fx will contain between them an odd number 
of roots of ^x, and the number of real roots oi fx greater than the greatest 
root of /«;, and the number of real roots oi fx less than the least root oi fx 
will each be even. Hence the effective intercalation scale between any two 
limits a and b will be formed by merely reducing the 77 groups to single 
units, and the number of h's in the scale so formed will be the total number 
of h's, between the limits a and b. Moreover, since such scale commences 
always with a root oi fx, or with an even number of roots oi fx followed by 



57] of two Algebraical Functions 521 

a root of fx, if the number of h's and ■jy's cut off be even, and with a root of 
f'x or an even number of roots of fx followed by a root of fx, if the number 
so cut off be odd, it follows that for this case I (a) - 1 (6), a being the 
superior limit, will be always positive, and will measure the total number 
of real roots of fx lying between a and b ; this, then, is Sturm's theorem, 
treated as a corollary to the Theory of Intercalations. 

Art. 51. If we write down the last syzygetic equation between fx of m 
and (px of n dimensions, namely 

it has been shown that the succession of signs in the series formed with/a;, 
^x and their successive Bezoutian secondaries will contain the same number 
of continuations and variations as the series formed with fx, t^-i {«'), and 
their successive Bezoutian secondaries. This indicates that the effective 
scale of interpositions for fx and </>« will contain an equal number of roots 
of fx with the etfective scale for fx and tm-i (;») ; the two scales however 
will not necessarily be identical, because the roots of <px will not necessarily 
be in the same order relative to the h's in the one scale as those of tm-\ («) 
relative to the A's in the other scale. This equality is perfectly well explained 
d posteriori by the form of <„j_i («), which by the formula in Section II. will 
be represented by 

^(x-h^(x-h^ (x-h ) #g,0/i,,-..</»/v„ 

Now, whenever x is indefinitely near to any one of the roots of fx, as hg^, 
this sum reduces to the simple expression 

0/i,, 0A,^ . . . <f>h,^_^ = {(/>Ai <j}L . . . 4>hJ^ -^ , 

and consequently in the immediate neighbourhood of every real root of fx, 
<f)X and tm-iix) will have always the same or always a contrary sign, 
according as <f>liq^(hhq„ ... (phq,^ is positive or negative, which will depend upon 
the relative disposition of the real roots in / and (f> ; in either case the 
effective" scale of interpositions for fx with (px and for fx with t,„^iX must 
contain the same number of h's ; but the difference will be, that if 
(phi^h^ ... (f>hm is positive an h will occupy the first place in each scale, or 
the second place in each scale ; but if negative, then in one scale an h 
will occupy the first place, and in the other scale the second place. 

Art. 52. The same process of common measure or residues which serves 
to furnish a rhizoristic series for fx or a syrrhizoristic series for fx and cpx, 
will serve also to furnish superior and inferior limits to the real roots of any 
proposed equation. Thus suppose fx to be any rational integral function of 



522 



On a Theory of the Syzygetic Relations 



[57 



X of the degree n and <^x any other function of x, which I shall begin 
with supposing to be of the degree (« — 1), and let the successive quotients 
resulting from the process of finding the greatest common measure of /«, 0« 
continued until the last remainder is not a constant but zero, be supposed 
to be (as they may generally be taken, but subject to cases of exception, 
which will hereafter be alluded to) n linear functions qi, q.2 ... qn', then we 
shall have 

^^ _ J_ J_ 1 1 

fx 5i + ^2 + " ' qn-1 + qn' 
and therefore 

4>x = KN, 

fx ^KD, 

where N is the numerator and D the denominator of the continued fraction 
and ^ is a constant ; the value of this constant is immaterial but is 
in fact 

- V U U 

Lo, Li, Li, L3, &c. being the leading coefficients of the last, the last but one, 
the last but two, &c. of the Bezoutiau secondaries tofx and ^x. Accordingly, 

if n = 1, let i) = gi = yu.j ; 

if w = 2, let D = ^j^i + 1 = /xj jgj + — I = /ii^2 ; 

if w = 3, let I* = 53 [q^q-^ + l}+qi = M1M2 ](?3 + — [ = /*i/^2/i^3 ; 



and in general let 
where 



i) = /ii/i„jU,3... fin, 



1 1,1' 

fj-i = qi, M2 = 2'o + — , /lis =5'3 + — >■■■ /i» = ?« + — ^• 

Now suppose X to be so taken that 

qi does not lie between + 1 and — 1' 

g-a „ „ + 2 and - 2 

qa „ „ + 2 and — 2 

Qi „ „ + 2 and — 2 f ' 



qn- 

qn 



2 and — 2 
1 and — 1 



(a,) 



where it will be observed that the excluded region lies between] + 2 and — 2 
for all the intermediate quotients, but between only + 1 and — 1 for the first 



57] of two Algebraical Functions. 523 

and last quotients. Then fi^ is positively or negatively greater than 1, 

therefore — is a positive or negative fraction ; but q^ is positively or nega- 

tively greater than 2 ; therefore /is will be of the same sign as q^, and also /Xj 

will be positively or negatively greater than 1 ; therefore — will be a positive 

or negative fraction; but q^ is positively or negatively greater tlian 2; 
therefore fx^ will be of the same sign as q^, and also fi^ will be positively 
or negatively greater than 1 ; and proceeding in this way, we find that all 
values of jjn, from i=l to i = n—\, will be of the same sign as qi, and 

positively or negatively greater than 1. Finally, will be a fraction, 

/^n-l 

and therefore, since g„ is positively or negatively greater than 1, /^n = 5'n + — 

h'n-i 

will have the same sign as q^ (but of course is not necessarily greater 
than 1, nor would that condition serve any purpose were it satisfied). We 
infer consequently, that when the conditions (a)) are satisfied, /tj, /igj 
yitj . . . (tt„ will respectively have the same signs as q-i, q«....qn\ and therefore 
I) = /u,iij,.2/u,3 ...fin has the same sign as qiq^qa ■■• q,i. Now suppose 

gi = Or^x + bi, q2 = n^x + h„...qn = a^x + h^, 

and solve the 2« equations 
aiX + 6i = + Ci, arfis + 62 = + Cg ... a^-i* + &k-i = c„_i, anX + &« = c,j, 
ai« + &i = — Ci, ajfl; + &2 = — C2 . . . a„_i« + &«_i = — c„_i , a„a;+6„ = — Cn, 

where 

Cj ^ X, Co ^ ^, Cg ^ ^ ... Cjj 1 =- ^j Cyi =— X. 

Whenever in any one of the n pairs of equations above written the coefficient 
of X is positive, the upper equation of the pair will bring out the greater 
value of x; but when the coefficient is negative the lower equation will give 
the greater value. Take the pair 

aiX +hi= Cj, 
aix + bi = — Ci, 

If ffli is positive aiX + bi will always be positive, and greater than Cj, between 
JT = 00 and X = the greater of the two values of a; ; if ai is negative a; a; + bi 
will always be negative, and less (that is nearer to — 00 ) than — Ci, for all 
values of x between the same limits as before. So again it will be seen 
in like manner, that whether ai be positive or negative, between a; = — 00 and 
a;=the lesser of the two values of x corresponding to the above pair of 
equations, aiX + bi will always retain the same sign, and will be greater than 
+ Ci, or less than — cj, according as a; is negative or positive. If, then, we 



524 On a Theory of the Syzygetic Relations [57 

take the greatest of the greaters of the n pairs of values of x, that is the 
absolute greatest of the 2n values, and the least of the lessers, that is the 
absolute least of the same, say L and A, then between L and A, 51, gj . . . 5^ will 
each always retain an invariable sign, and will then fall without the limits 
+ Ci, ± C2, ... + c„_i, + c„, so that between + 00 and L and between A and — oo , 
fM-^/j,^... fXn, that is a constant multiple of /(«), will retain the same sign as 
5,52 ■■■qn, that is will never change its sign from the beginning to the end 
of one interval, nor from the beginning to the end of the other; and con- 
sequently L and A will be a superior and inferior limit respectively to the 
real roots of fx. It will of course be observed that it is indifferent for the 

purposes of the foregoing theorem, whether ^ be expanded under the form 

of a proper or an improper fraction, that is whether we employ the ordinary 
or the Sturmian process of successive division ; for changing the signs of the 
residues will only have the effect of changing qi into {±)qi, and the pair 
of equations {±)qi= ± Cj remains the same whether the + or the - sign be 
prefixed to qi. The result is, that if we form the 2n quantities 

+ I-&1 ±2-b, ±2-b, +2- bn-i ±l-bn 



the greatest of them will be a superior, and the least of them an inferior 
limit to the roots offx*. 

It may be remarked that if the successive dividends in the course of 

the process be multiplied respectively by k-^, k^ ... ^"„, ~ will take the form 

iCi 1C2 i^z "^n 

qi + q2 + qi + "' qn 

and if we write 

a^X + 61 = + Ci, a^ + &.J = + Ca . . . QnHC + 6,1 = + Cn 

and make 

Ci = l, c2=\+k^, C3 = I + ^3 ... c„ = 1 -l-A.',i, 

the same reasoning as above will show that the greatest and least of the 2m 
quantities 

+ 1-61 + (1 + k^) - 6a ± (1 + kn) - bn-i ± I - bn 

will be a superior and inferior limit to the roots o{fx. 

For greater simplicity, again, consider k^, k.^... kn to be all equal to unity; 
we may make this addition to the theorem as above stated, namely calling 

* For a generalization and improved form of statement of this theorem see Supplement to 
the present Section. 



57] of tivo Algebraical Functions. 525 

Zi, A]) ^2. Aj ... Ln, A„ the greatest and least values of the terms contained 
respectively in the series marked below 1, 2, 3 ... n, namely — 

±l-h, ±^-h ±2-63 



+ 1 



(fa 




± 2- 


■h 


tts 




± 1- 


h 



±2- 


- hn-i 


a« 


— 1 


±2- 


■6„_i 


a«-i 


±2- 


■&/1-1 



±1- 


hn 


a„ 




fl- 


■hn 


ail 




±1- 


■K 



+ 1 - 6„_, + 1 - 6„ 



+ l-6„ 



(1) 
(2) 
(3) 

(n-1) 
(n) 



ii, Ajj ij, Aj ... Z„, A„ will be respectively superior and inferior limits tofx, 
<^x and their successive residues. As a corollary, we see, of course, that L 
and A, the superior and inferior limits to the roots of the given function fx, 
must always lie between + 00 and the greatest root, and between — 00 and 
the least root, of the arbitrarily assumed function ^x. 

Art. 53. Let us now assume somewhat more generally that <px is any 
number of degrees 6^ in x lower than fx, which will cause the first quotient 
g'e, to be of the degree ^1 in a; ; and let us further suppose that (f)X stands in 
such a relation to fx that the following quotients, q^^, qg^... qg , are of the 
degrees 6^,63... dp in x {O^.d^.-.O^ being supposed not necessarily units, 
as they would generally be, but any positive integers whatever, as may 
happen in consequence of one or more of the leading coefificients in any 
residue vanishing) ; then 

^ _2 1 ]_ J^ 

fx~ qe, + q6, + qe,+ "' ?«/ 

where 6-^ + 6^+ 6^+ ... + O^ — n; and consequently /a; will be equal to the 
denominator of the last convergent above written, multiplied b}^ a constant, 
so that we have now cfx = nii J?i2 • • • w-p, where 

1 _ 1 

//tj "'■p — 1 

And as in the case previously considered, so long as 

>1\ />2\ />2\ />1 



ffe, 1 or 1 , qs^ 

v< - 1/ \< - 2/ \< 

fx will have the same sign as qe,qe,...qe 



526 On a Theory of the Syzygetic Relations [57 

Let now g'sj = + Ci, 9e, = + C3 . . . q^^ = + c^, 

where Ci = 1, 0^ = 2 ... Cp_i =2, Cp = 1. 

Consider any pair of the above equations as q^f — Cj^ = 0. 

Firstly, suppose all the roots of this equation are impossible ; q^^ — cf 
must be positive for all values of x, and q^. can never lie between + Cj and 
— d ; moreover, since upon the hypothesis made, q^^ + Cj and q^. — Cj always 
retain the same sign, namely, that of the coefficient of the highest power of 
q^., it follows that q^. must also always retain the same sign ; for if we con- 
struct the two curves y = ^e. + Ci and y = qg.— Gi, these will both lie on the 
same side of the axis of x, and never cut the axis, consequently the curve 
y = qBi, which lies between them, must also lie on the same side as either of 
them, and never cut the axis. 

Hence, then, if the roots of the equation are all impossible, qg. will 
always retain the same sign, and will never fall within the region bounded 
on two sides by + Ci and — Ci. 

Secondly, suppose the equation to have one or more possible roots, and 
li to be the greatest, and X; the least (which of course, if there is but one 
possible root, will be identical). If the leading coefficient of qg. is positive, 
the greatest root (I) of the equation q^. — Cj = will exceed the greatest root 
(l') of the equation qe. + Ci = 0; for between x = 00 and x = I', q^^ must go 
through all values intermediate between 00 and — Cj ; hence there must be a 
quality I intermediate between I' and + oc , which will make q^^ = d. In 
like manner, if the leading coefficient of g'^j is negative, it will be seen that 
the greatest root of g^j + Cj = will exceed that of g^i — Ci = 0. Moreover, 
in the one case g^. will be always positive and greater than d, and in the 
other always negative and less than Cj. In every case, therefore, between 
+ 00 and li, qe- retains the same sign, and does not fall within the region 
bounded by + Cj and — Cj ; the same thing may be shown to be true for 
all values of x between — 00 and X;. Hence, then, by the same reasoning as 
that employed in the preceding article, we are enabled to affirm, that if we 
form the equation 

(qe,' - 1) (qe: - 4) (g<,/ - 4) . . . (g^. - 4) (V - 1) = 0, Or) 

its greatest root will be a superior limit, and its least root an inferior limit 
to the roots of the equation fx = 0, whatever be the value of the assumed 
function (j)x ; and if the above e(iuation (t/t) has no real root, all the roots of 
fx will be imaginary. 

Art. 54. In the preceding two articles it has been supposed that all the 
quotients are taken integral functions of x; but the process of successive 
division may be so conducted as to give rise to quotients of the form 

ax'' + bx^-^+ ... + C + - + ... + — . 



57] of two Algebraical Functions. 527 

Suppose then that we have in general 

^x \ \ 1 , 

"where q^, q^.-.q^ are each of the general form above vsrritten (but of course 
i and i' being not necessarily the same for any tvfo of the quotients), and 
suppose that the sum of the degrees in x of q^, qz...q^ is n+t, where t is 
essentially (as it must be) positive. Then we shall find, as in the last article, 
that L and A being called the greatest and least roots of 

(?i=-l)(g/-4)...(g-^„-:-4)(g„^-l), 

D, the denominator of the last convergent to the continued fraction above 
written, will never change its sign between + oo and L, nor between A and 
—00 ; but here we shall have 

fx = Kx* X D. 

Hence x^D will be invariable in sign within each of these two intervals. 

Firstly, let t be even ; then fx will be invariable in sign, whatever L 
and A may be for each such interval. 

Secondly, let t be odd ; then if X is > and A < 0, fx cannot change 
its sign in either interval ; but if i is < or A > 0, _/a; will change its sign as 
X passes through zero, but will be invariable for each of the three regions 
contained between + co and L, L and 0, or and A (as the case may be), 
and A and — czj ; so that universally L and A will be a superior and inferior 
limit to the roots oi fx, making abstraction of the roots (if any such there be 
vnfx) whose value is zero. 

Art. 55. I shall close this section with offering (for what it is worth) 
a bare suggestion as to the mode in which the theory of Intercalations may 
hereafter be found to admit of being extended from a system of two general 
functions of x, to a system of three general functions of x, y, four general 
functions of x, y, z, and in general to a system of e general functions of e — 1 
variables, or which is the same thing, of e homogeneous functions of e 
variables. In the case of two functions of x, fx and <^x, fx = and (f>x = 
may be considered to represent two systems of points in a right line ; and 
the theory relates in this case to the relative positions of these two 
" Kenothemes " or point systems ; and of course using x and y to denote the 
distances of any point in a line from two fixed points therein respectively, 
instead of fx and (px, we may employ two homogeneous functions of x and y, 
as f{x, y) and (f) {x, y), to denote these two systems of points. So, similarly, 
if we have three functions of two variables, f(x, y), g (x, y), h(x, y), which 
I shall suppose to be of the same degree, we may consider the mutual 
relations of the Monothemes, that is to say, the three plane curves, denoted 



528 On a Theory of the Syzygetic Relations [57 

by the equations f{x, y) = Q, g (x, y) = 0, h (x, y) = 0. Now every two of 
these will intersect one another in a system of points, which we may call 
(/. 9) foi' tb® intersections of/ and g, {g, h) for those of g and /i, and (h,f) 
for those of h and f. If we take any two of these systems of intersections, 
as (/, g) and (g, h), they will both lie upon one of the given curves (g). 
And by reading off the two systems of points (/, g) and (g, h), arranged 
according to the order upon which they are disposed upon the curve g, we 
may, by following the course of such curve, form a scale of effective inter- 
calations for these two systems, and in like manner for the two systems 
(g, h) and (A,/) ; Qhf) and (/, ^f). Now I believe that it will be found that 
when/ g, h represent any algebraical curves consisting of a single continuous 
line, either extending to infinity in both directions, or returning to itself 
(and I have fully satisfied myself of the truth of this for the case of ellipses), 
each effective scale of intercalation will contain the same number of pairs of 
points ; if, however, the curves consist of more than one branch, as if hyper- 
bolae be considered, such is no longer necessarily the case ; from these facts, 
conjoined with the light thrown upon the subject by its relation to the 
theory of combinants explained in the succeeding section, I am induced to 
infer the probability of the truth of the following law (which, for avoidance 
of further uncertainty, I confine to the case of functions of the same degree), 
namely, that if /, g, h be three homogeneous functions of x, y, and s of the 
same degree, and if U, V, W be any three linear functions of/ g, h, and if 
U = 0, V—0, W = be treated as the equations to three cones, and if we 
form an effective scale of the intercalations of the lines of intersection of U 
and W, and V and W, according to the order in which they are disposed 
upon W (which seems to require that the lines shall be continuous, in order 
to admit of a fixed order of reading off the intersections of any two of them 
upon the third) ; Then, whatever value may have been given to the coeffi- 
cients in the linear functions, the number of elements remaining in any such 
scale will (as I conjecture) be constant, and some theory (to be discovered) 
for three functions, analogous to that of Bezoutian residues for two functions, 
will serve to determine the number of the elements so remaining. And so, 
in like manner, but with a difficulty increasing at each step (as at the next 
step we should have to pass into quasi-space of four dimensions), a theory of 
intercalations may be conjectured to exist for any ?i general functions of 
any {11 — 1) variables. 

Development of the method of assigning a superior and inferior limit 
to the roots of any algebraical equation. 

Art. (a). Since the articles in the preceding part of this section on the 
method of discovering limits to the roots of an algebraical equation were 
written, the method of which the germ is therein contained has presented 



57] of two Algebraical Functions. 529 

itself in a much more fully developed form, which I proceed to exhibit: for 
greater simplicity I shall suppose <^x to be of n — \, and fx to be of n 
dimensions in x, and that by means of the ordinary process for common 
measure (except that as in Sturm's theorem the signs of all the remainders 
are changed) ^ has been thrown under the form of the improper continued 

fraction 

1 J_ J_ 1^ 

qi-q2-qs-"' qn' 

where qi, q^ ...qn are all restricted to signify simple linear functions of x. 

Suppose the series qi, q^, q^... ^n to be resolved into the distinct sequences 
qiq^-.-qu qi+^qi+2...qi', gi'+i ... g^. ••-.gra+i ••• ?», 
in such a manner that in each sequence, as g^+i, qi+^...qi; the coefficients 
of X have all the same sign, but that in any two adjoining sequences the 
coefficients of x have opposite signs, so that for instance in q^ and qi+j^ the 
coefficients of x are unlike, as also in qi' and q^+i ; there will of course be 
nothing to preclude any of these sequences becoming reduced to a single term. 

The first theorem is, that the greatest and least roots of the product of 
the cumulants [p. 504 above] 

[q,q„. ...qi]x [qi+iqi+. ... qi'] ... x [qi{,+^q^i^+, ... g,J 
are superior and inferior limits to the roots of fx. To prove this theorem. 
I begin with premising the two following lemmas, one virtually and the 
other expressly contained in the Philosophical Magazine for the months of 
September and October of the present year* [p. 641 below]. 

* Each of these two lemmata flows readily from the faculty previously adverted to engaged 
by every cumulant of being representable under the form of a determinant. As to the second 
lemma, it becomes apparent immediately when the cumulant is so represented, by separating the 
matrix into two rectangles and expressing the entire determinant according to a well-known rule- 
for the decomposition of determinants as a function of the determinants belonging to these two- 
rectangles taken separately. As to the first lemma, by reason of the cumulant [wj oi„... wj_j «jWj-j.i J 
being so representable, we know that when [wju, ... Wj-_i(<j,-] = 0, [uj u, . . . Uj_i] and [wi Wj . . . 0^+ J 
must have opposite signs. Suppose, now, that the theorem is true when the number of elements, 
in the type does not exceed i; then the roots of [uj wo... Uj._j], say of i/'j-j, being called 
\,li2...'lii-i, and of [wioij ... «j_iuj, say of f^, being called \,'k^...ki, these may be 
arranged in the following order of magnitude k^, \, k^, h^, k^ ... ij-i, /jj_i, hi ; and if the roots, 
of [U1W2 ... Wj-iWiWi+i]. say of fij^-^, be called l^, l^ ... J,-^.j, from the fact of the leading coefficients 
in \pi_-^ and \j/ij^^ expanded according to the powers of x having the same sign, it follows that 
when x=oo, ;frj_j and 1//;+! have the same sign, but they have contrary signs when x=k^; but 
fi-j^ does not change its sign between x = oo and x = k-^, hence fi^-^ does change its sign between 
a;=<x> and x = k-^, and therefore a root of i/-;+j lies between co and k^; in like manner precisely it 
may be shown that a root of ^j+j lies between -00 and fcj-; and since i/-,-.! changes its sign 
between A-j and \, between k^ and k^... and between fcj_i and ft,-, ^i.^.^ must likewise change 
its sign between one and the other extremity of each of these intervals, and hence the roots 
ij, Z3 ... Zj+i are intercalated between od , fti, fcj ... A^, - cc , or which is the same thing, k-^, k2...ki 
are respectively intercalated between l^, l^ ... Z,-+i; consequently, if the theorem is true up to i, 
it is true for i + 1, and therefore true universally; but is manifestly true when i = 2, for then 
a;=±oD makes [uiUo], that is, UjUj-l positive; but ui = makes it negative, which proves. 
the theorem contained in Lemma A. 

s. 34 



530 On a Theory of the Syzygetic Relations [57 

Lemma A. The roots of the cumulant \(jiq2...qi\ in which each element 
is a linear function of x, and wherein the coefficient of x for each element 
has the like sign, are all real, and between every two of such roots is con- 
tained a root of the cumulant \<liq2 ■ ■ ■ qi-i\, and ex converso a root of the 
cumulant [^'aS's ••• g'J; and (as an evident corollary) for all values of p and p' 
intermediate between 1 and i the greatest root of [qiq^ ■ ■ ■ qi-iqi] will be 
greater, and the least root of the same will be less, than the greatest and 
least roots respectively of [qpqp+i ... qp'-iq^']. 

Lemma B. For all values of the elements q^q^.-.q^, the cumulant 
{q, q.... q^^^ q^ q^+, q^+o . . . ^„] ^[qiq,... g.-i q^ ] x [q^+, q^^, . . . g„] * 

-[qiq-2---q.-i] x [q^+. ■ ■ ■ qnl 
Thus for example the cumulant [abed], that is 

abed - ab — cd — ad+l = [ab] x [cd] — [a] x [d] = (ab — 1) (cd — 1) — ad, 
and [abode], that is 

abode — abc — abe — ade — ode + a + c + e = [abc] [de] — [ab] [e], 
that is = {abc — a — c){de — 1) — (ab — 1) e. 

Art. (/3). Also suppose that gi^a .•• q,^qa+i ■■■ % are all linear functions of 
X, and that the coefficients of x have all one (say the positive) sign in 
§1,^2 ••.5'a, J and all the contrary signs in g^+j ... qn, and let L be not less than 
the greatest root of [qiq^.-.q^ or of [q^+i ■ ■ ■ qr^, and also let A be not 
greater that the least root of each of these same two cumulants ; then by 
Lemma A, L and A will also be respectively greater than the greatest, and 
less than the least roots of [qiq^ ... ?«,-i] and of [g„+2 ... g'J- Now the coeffi- 
cient of the highest power of x in both [q^^qi. . . . q^] and in [q^q^... q^^^ is 
positive, but as to [q^+i •■■ g'w] and [q^+n ... g„] is of contrary signs in the two, 
namely, negative in that one of those cumulants which contains an odd, and 
positive in that one of the two which contains an even number of elements. 
Hence by virtue of Lemma B, L and any quantity greater than L substituted 
for X will make [qiq^. ... 5m] to have always the same sign, and in like manner 
it may be shown that A and any quantity less than A substituted for x will 
also cause [^iga ... g^] to retain always the same sign. Hence L and A are 
superior and inferior limits to [qiq^-'-qn]', and the same reasoning would 
evidently apply if we had supposed the signs of the coefficients of x in the 
first partial series of elements to have been negative, and in the other series 
of elements to have been positive. 

The greatest and least roots of [qiq^ ■■■ ?„] x [q^+i ... qn] evidently satisfy 
the condition to which L and A are subject, and may be taken in place of L 
and A respectively. They will accordingly be superior and inferior limits to 
the cumulant 

[qiq^....q^q^+i...qn]- 



57] of two Algebraical Functions. 531 

Again, by virtue of Lemma B it may readily be shown that 
[51 go ... 5„., g..+ig„,+2 ... go,,, 9«„+i •■• 2«] 

= [g-Ws • ■ • g. J X [g„,+i5„j+2 . . . g„J X [g„^+i . . . g„] 

- [^1^2 • • . g<a.-i] X [g„j+2 . . . g J X [g,^+i . . . g„] 

- [gig, . . . g„ J x [g„^+i . . . g„^^J x [g„^+2 . . . g„] 
+ [gi ga . . . g„,-i] x [g„,+2 . . . g„^_x] x [g„^+2 . . . g„] ; 

and hence if gi, q^.-.q^ are all linear functions of a; in which the coefficients 
of X have all the same algebraical sign in any one (taken 'per se) of the three 
series 

2ig2-..g»i, g„,+i...g«,,, g,.,+i...g„, 

but so that this sign changes in passing from one series to another, it is 
easily seen, by the same reasoning as in the preceding case, that the two 
positive and two negative products on the right-hand side of the equation 
all give the same sign to the coefficient of the highest power of x, and 
consequently that if L and A be superior and inferior limits to 

[?!••• ?-,]. [?a„+i • • • ?».]- [?»,+! • • • qnl 
and consequently by Lemma A, to 

[q,q, . . . g„,_J, [g„,+, . . . g„J, [q^^+, . . . q^^_{\, [g„,+„ . . . g„^_i], 
and to [g<o,+2---g»i]. 

X or A substituted for a; will cause {(^iq^ ■■•q7i\ to retain always the same sign, 
and will consequently be superior and inferior limits thereto ; and so in 
general ; whence it follows, returning to the theorem to be demonstrated, 
that the greatest and least roots of 

[gig2 ... gi] x [gi+igi+2 ... gi] x ... x [gH,+i ... g„], 

will be superior and inferior limits to the cumulant [gig2... gj, that is to 
Gfx*, and therefore tofx, as was to be proved. 

Art. (7). The second theorem is the following: if gi,g2...gm be linear 
functions of x, say a-i_x + 61, a„x + &„•■. o,nX + 6„, in which the coefficients of x 

* If ^ expanded as a continued fraction by means of the common measure process gives 
jx 

rise to tlie quotients Ji, Sa ■•■ ?»' ^^^ i^ L-^, L^ ... i„_i, i„ be the leading coefficients of the 
successive simplified residues, (i„ being, in fact, the final simplified residue, that is, the 
resultant to (px, /a;), we must have tpx = C\_q^, q^ ... g,J, fx = C[q-y, q^... g„], where (supposing (px 
to be of ra - 1, and fx of n dimensions in x), 

1 \ L^L\_^L\_i (fee. ) 

34—2 



532 On a Theory of the Syzygetic Relations [57 

have all the same sign, and if we take the quantities fj^, /u.^... fin-i, all 
having the same sign as aj, as ... a™, but otherwise arbitrary, and make 

ki = fli, k2 = fl2-\ . ^3 = A'3H . . . ifcn-i = /in-i H , kn=-——, 

then the greatest of the quantities 

ki — &i ^2 — bo kn — hn 



say L, is a superior limit, and the least of the quantities 

— ki—h-^ —k.2 — b2 —kn — hn 
Ox ' a^ '" an 

say A, is an inferior limit to the roots oifx. 

L and any value greater than L substituted for x will evidently make 
<j[i — ^i> q^—k^... qn — kn, all of them positive. 

Hence, when a; = or >L, q^ is positive and > jMi, and 

q„ >k« >WoH , that is, is positive, and > Ua, 

n„ >k, >M3H > tbat is, is positive, and > yUs, 



and Qn . . . — > , that is, is positive, 

qn-x-qn-1. 9i M»-I M«-i 

and consequently the cumulant [^i^a^'s ••• ?«], which 

= ^.x(^.-^jx(g3-^^jx&c., 

remains of a constant sign when L and any quantity greater than L is 
substituted for x. Hence Z is a superior limit. In like manner A and any 
quantity less than A will evidently make gi + ^"i, go + fc ... qn-\-kn all of them 
negative, so that, when « = or < A, gj is negative, and < — /^i, 

q„ < fe is negative, and < — tt,, 

gi /^i 

o, < ^^3 is negative, and < — itj, 

^ g2 /"2 '' 



1111 1 . 
and ff„ . . . - < is negative. 

qn-\-qn-i qi ^n-i IJ-n-i 



57] of two Algebraical Functions. 533 

So that [g'l , q.^... q^ for all values of x less than A will preserve an invariable 
sign, and consequently A is an inferior limit tofx. 

Art. (8). It may be remarked that the quantities 

1 1 1 11 

/*! A'a Ml— 3 Mn— 2 Mm— 1 

may be derived successively from one another, according to the same law, 
from whichever end of the series we begin. 

If we take any two consecutive terms as 

1 1 

Mi + > Mi+i + — , 

Mi-i Mi 

the effect of diminishing fii is to decrease the first of these two terms, and 

pro tanto, to tend to reduce the limit; but on the other hand, — being 

increased, there is brought into play an opposite tendency, which operates 
pro tanto to increase the value of the limit. 

Art. (e). It is of importance to remark, that by a right selection of 
the system of quantities /ij, /x2--.m«-i. which enter into the composition of 
k-i, Jc2...hn, L may be made to coincide with the greatest root of [g-i, q^.,. g,,]; 
and so in like manner by a right selection of another system of these 
quantities, whereby to form k^, k^.-.kn, A may be made to coincide with the 
least root of the same. Thus let fj^, fi^... fin-i be so chosen, that 

q^-ki = 0, q2-ki = ...qn — kn = 0, 

are all satisfied by the same value of x. 

Ill 

Then g'i = Mi. 9'2 = M2H — . g'3 = M3 + — ... 9« = , 

Ml M2 M"— 1 

exist simultaneously. 

XX 1 111 

Hence M2 = 22--, ^3 = ?3--7 = ?3--— -, 

qi M2 i2~ vi 



Mn— 1 — ?m— 1 " 



1 



5»— 2 ~ 9^n— 3 S'l 

111 

M» = ■ ■ • r ' 

?»— 1 ~ S'l-s ~ 2i 

which is satisfied by making 

[g«, qn-i, qn-i--- 2i] = 0. 
It remains then only to show that the greatest root of x in this equation 
substituted for x m. q^,q^...qn will make p.^, fi.2... fin-i all of one sign, and 
that the least root of x similarly substituted, will also make them all of one, 
but a contrary sign, which may be proved as follows. 



534 On a Theory of the Syzygetic Relations [57 

We have 

/ti = qi, ^2 = [qi, ga] -^ gx, /^s = hiq^^qs] ^ [qi, q^] &c. 

' /^«-i = [?! 9'2 ■ ■ ■ gn-i] -^ [g'l ?2 • • • g>i-2] ; 

and by Lemma B the superior limit to [^ig, ■•• ?«] will be a superior limit 
also to [^1^2 ••• 9'n-2], and to 

{qiq^l fe?2?3]..-, [g-igs ••■?«-!]. 
Consequently this superior limit will make yUi, /Xj ••• /in-i have all the same 
sign as that of the coefficients oi x\n q^, q^.-.qn- And in like manner, the 
inferior limit to [qiq^ ... 5™] will cause ji^, ^2 ■•■f^n-i to have all the contrary 
sign to that of these coefficients. 

Thus then we see that when the coefficients of x in the partial quotients 

to ^ expressed as an improper continued fraction form a single series of 

continuations of signs, by a right choice of the arbitrary constants /ii,/x2...yu-„_i 
the superior or inferior limit given by this new method may severally and 
separately be made to coincide with the greatest and least real root, or each 
in turn with the sole real root offx, if there be but one. 

Art. (^). The general method of enclosing the roots oi fx within limits 
is founded upon the combination of the two theorems above demonstrated. 
An arbitrary function (j}x, one degree in x below fx, being assumed, and by 
aid of the auxiliary function (f)X, fx being thrown under the form 

C[q,q._...qi, q^'ql . . . q'f, ql' . . . (qUq)^ . . ., (q)({,], 

in which the coefficient of x is supposed to change sign in the passage from 
qi to 5/, from q'i', to q", &c., a superior limit is found to each of the 
cumulants 

[qiq^ ■ ■ ■ qil [qi'q^ ■ ■ ■ gV] . ■ . [iq\{q)2 ■ ■ ■ (?)©], 

taken separately, by means of the second theorem, and then by virtue of the 
first theorem the greatest of these superior limits is a superior limit to the 
cumulant 

feg2...g'i...(g)i...(?)(i)], 

and consequently to fx, and so mutatis mutandis the least of the inferior 
limits of the same partial cumulants is an inferior limit to the total cumulant 

[q,q^...qi...{q\{q\...{q)(t,l 

Art. {t)). When all the roots of fx are real, if ^x be so assumed that all 

its roots are intercalated between those oi fx, the partial quotients to ~ 

will form but one single series. In order that ^x may fulfil this condition, 
it is necessary that the coefficients of <^x shall be subject to certain conditions 



57] of two Algebraical Functions. 535 

of inequalit}'^, not necessary to be investigated here ; but no conditions of 
equality, that is, no equations between the coefficients of <^x, are introduced 
by this condition; or in other words, the coefficients* of (/>«, the auxiliary 
function, are independent and arbitrary within limits ; and we have shown 
that in this case the auxiliary constants fi^, fi., ■■■ f^n-i may be so determined 
that the limits may be made to come separately and respectively into 
contact with the two extreme roots. When all the roots of fx are not real, 
the quotients (however (px is chosen) can no longer be made to form a single 
series. It still however remains true, that, by a due choice of the auxiliary 
function followed by a due choice of the auxiliary constants, this coincidence 
may be brought about, so long as there is a single real root infx. 

It is rather important to demonstrate this universal possibility of 
effecting a coincidence of the limits to the roots with the extreme roots 
themselves, because it is the most striking feature which distinguishes the 
method of limitation here developed from all others previously brought to 
light. 

Art. (6). Before entering upon this demonstration I may make the 
passing remark, that every method of root-limitation is implicitly a method 
of root-approximation. 

For instance, let e be any given quantity between which and -!- ao it is 
known that a root of fx lies. Then if we write x = e + -, and form the 

J y 

equation y'^f[e + -\ = 0, and find L a superior limit to y, it is clear that 

e + Y will lie between e and the root of fx say E, next superior to e. Again, 

making x = e + y + -,, and finding a superior limit L' to y', we shall have 

e + y + y, still nearer to E than e + y was ; and so we may proceed advanc- 
ing nearer and nearer, and always from *:ae same side towards E at each 
step, and finally obtain E under the form e -)- j -f j; -I- -p, +&c. And in like 
manner calling E, the root next below e, we may find 

E,=e--^ --r-,--r7/, &C. 

' AAA 

Art. (t). In establishing the theorem of coincidence above adverted to, 
the following notation will be found very advantageous. Let XI denote 
a Type of any number of Elements, as ^i, q^... qi-i, qi, and let CI' denote this 

* It need scarcely be stated that fx is the simplest form of <px, which satisfies the condition 
in question. 



536 On a Theory of the Syzygetic Relations [57 

same type when the last element, and 'fl the same type when the first element 
is cut off, and 'fl' the same type when both extremes are cut off, so that the 
apocopated type 12' will mean q-i, q.2...qi-i', the apocopated type 'O will 
mean q„q3 ...qi, and the doubly apocopated type 'fl' will mean q„^, q-^ ... qi-i. 

If now a type O be made up of the types fij, fij ... Hj put in apposition, 
and if we use in general [XI] to denote the cumulant corresponding to the 
type XI, there will be a very simple law* connecting [X2] with 

[n^], [XI,], [XI3] . . . [Hi.,], [Hi.,], [XI,], 

[ii\l[D.\i[ir,]...[ii'i.,l[ii'i-.l 

['XlJ,['Xl,]...['Xl,_J,['Xi;_,],['XlJ, 

This law will be seen to be obviously deducible by successive steps of 
expansion from the fundamental theorem given in Lemma B, Art. (a), for 
the case of X2 = XljXi,, and will be best understood by showing its operation 
in a few simple cases. 

Thus let XI = XliXlsf. Then 

[XI] = [Oi] X [Xi,] - [Xl'i] X ['HJ. 
Let Xi=XiiXl2n3. Then 
[X2] = [nj X [XIJ X [XI3] - [Xl'i] X ['XIJ X [X23] - [Xli] X [Xl'J X ['£!,] 

+ [D,\] X ['XI',] X ['Xls]. 
Let XI = X2i XI2 XI3 XI4. Then 

[n] = [xijx[xijx[xi3]x[xij 

- [Xl'J X ['XI,] X [XI3] X [XI4] - [Oi] X [Xl'J X ['Xlj] X [XI4] 

- [XI,] X [nj X [a'3] X ['xijt + [a\] X ['Xl'J X ['X23] X [XIJ 

+ [Xi'J X ['X}J X [Xi'3] X ['XIJ + [Xi J X [Xl'J X ['Xl'3] X ['X2J 

- [Xl'J X ['Xl'J X ['Xl'J X ['XIJ, 

* The cumulant corresponding to any portion or fragment of a type may be said to be 
a partial cumulant to the entire type, and a type whose elements are constituted out of the 
elements of two or more types placed in juxtaposition may be said to be the aggregate of these 
types ; the law given in the text above may then be said to have for its object the expansion of 
the complete cumulant to any type in terms of complete and partial cumulants to the types 
of which the given type is the aggregate. 

t The sign of equality is employed here to denote the relation between a concrete whole and 
the aggregate of its parts. 

t The number of distinct factors entering into these products, taken collectively, is evidently 
•i + 2 (i - 1) + (i - 2), that is 4 (i - 1). 



57] of tivo Algebraical Functions. 537 

and so in general if fl = flifij ... Hj, [fl] may be expanded under the form 
of the sum of 2*~' products separable into i alternately positive and negative 
groups containing respectively 1, {i — 1), \ (i — 1) {i — 2), ... (i— 1), 1 products. 

Art. (k). In every one of the above groups forming a product the accents 
enter in pairs and between contiguous factors, it being a condition that if 
any fl have an accent on the right the next il must have one on the left, 
and if it have one on the left the preceding D, must have an accent on the 
right, and the number of pairs of accents goes on increasing in each group 
from to i—1. This rule serves completely to define the development in 
question *. 

For greater brevity let [flj, [Xl'J, ['ii«], ['fJ'J be denoted respectively 
by (Oe, co'e, 'coe, '(o'e, then when the type il^ consists of a single element, 

It should be observed that the two equations eog = 0, co'e = cannot exist 
simultaneously, for if fig represent q^, q^ ...qi, 

(Oe = qiw'e — &)"(., Ctj'e = qi-i(o" e — £<)"'e, &C., 

so that if (Og = and m'e = 0, we have w'e = 0, a)"'e = 0, &c., and thus, finally, 
— 1=0, which is absurd. 

Now, if we suppose Xlj, n^-.-ile to be types every element in each of 
which is a linear function of tv, the coefficients of so in these elements being 
positive in Oj, negative in fl^, and so on alternately, and fl is the aggregate 
of Hi, O2 ... Oe, it may easily be made out that each term in the development 
of CO in terms of Wj, co'i, 'toi, 'co\; co^, <o\, '<o^, ' <o\, &c. will have the same sign 
when we give to a; a value which is a superior limit, or an inferior limit to 

* When each partial type consists of a single element, every doubly accented fi will vanish, 
and every singly accented will become unity ; hence we may derive the rule for the expansion 
of the cumulant \a^a„a,^ ... a,] in terms of a^ , a^ ... a^, which will accordingly consist of 

a,a„ao ... flj-S (a,ar,...a,) + S (o,a2---''i)=F&e-, 

the indices e and /, e + 1 and /, &c. being understood to be all distinct integers (which agrees 
with the known rule for the expression of the denominator of a continued fraction in terms of 
the quotients). The number of terms in this expansion, in consequence of the vanishing of the 
quantities affected with a double accent, reduces from 2»'~^ down to the tth term in the series 
commencing with 1, 2, 3, &c. defined by the equation w,-^j=Mj + «.i_i, that is 

s/5\ 2 J J5\ 2 J ' 

the number, therefore, of products in which double accents occur in the general expansion of 
[wiwo — ujis 



538 On a Theory of the Syzygetic Relations [57 

the roots of each of the cumulants Wj, &)„...(»«, and consequently to those 
of the cumulants (o\, a)\ ... oo'e ; 'coj, 'u^ ... 'coe', 'w'l, 'w'j ... 'to'e ; the products 
affected with positive signs being all positive or negative in themselves, and 
those affected with negative signs being reversely all negative, or all positive. 

Thus, for example, if 

(o = a>i(On— co\ '0)2, 

and the sign of the leading coefficient in 'o), will be the contrary of that 
in £02, but (Ml and q)\ have both the same positive sign ; so again if 

ft) = ajift)2ft>3 — ft) 1 (Ooco^ — &)ift)Vft)3 + ci)\'co^ o}^, 

where the leading coefficients in o), and 'm^ have contrary signs, as have also 
those in as^ and co\ while w^ and 'ai\ have the same sign ; and of course 
the leading coefficients in &)i, a>^, w\, 'a>s have all the same sign, they being 
all positive, and so in general. But the superior limit to the roots of any 
integral algebraical function of x substituted in place of a; causes the signs 
of the resulting values of the functions to coincide with the signs of the 
leading coefficients, so that in the example last above given, L a superior 
limit to all the factors in the several products in the equation substituted 
for 00 will make 0)10)20)3, — co'/ai^aya, — &>iO)V«s, coi'co'/ms to have all the same 
sign. The like will be true of A the inferior limit; for if Hi, Oo, O3 contain 
respectively Wi, n„, n^ elements, the values of the four products last above 
written, when « = — 00 , will be to the values of the same when « = + 00 in 
the respective ratios of 

and so in general. Hence we deduce the theorem, that if the total type £1 
represent the aggregate in apposition of the partial orders D.^ , D,^ . . . D,^ (the 
elements being understood to be linear functions of x, Avhich are subject 
to the law of alternation in the signs of the coefficients of x in passing from 
one partial type to another), no superior limit to 0)1, 0)5... ft)^ can make <o 
vanish unless each separate product in the expansion of a> in terms of 
o)i, 0)2 ... ft)e and the appurtenant apocopated cumulants vanish separately. 

Art. (X). From the above theorem we may deduce the following law, 
namely, that if the roots of o)i, 0)2 ... o),, be supposed to be arranged in order 
of magnitude, and X to be that one of them which is nearest to + 00 or to 
— 00 , then if e is even it is impossible for X to be a root of co. Thus suppose 
e = 2, and consequently o) = o)i ft)^ — co\ 'co^ ; if X be a root of co^ and one of the 
two extremes of the roots of (o^, 0)2 put in order of magnitude, X cannot be 
a root of '0)2, for the roots of 'to., are confined between the roots of o),; but 



57] of tioo Algebraical Functions. 539 

if X make a and wi each vanish, we must have a)\ 'a^ = 0, hence &>/ = 
as well as (»i = 0, which is impossible. In like manner if a root of w^ were 
the extreme root, the same impossibility could be in like manner established. 

Again, suppose e = 4, so that 

(, ft)V«2 (bV&'s '»'s''»4 , co\'co\'(Os ay'ia^a'sOJi 

(O = tUiMoCUoWj il 1- 1 

[ a)iC02 coryoit a>sCOi tyiCOjWj eoiOJiCo^cOi 



Let X continue to denote one or the other extreme of the roots of 
a)](B„tt)3C()4. If X makes o) = we have 

Ml(U2(B3&)4 = 0, (»V'^2'^3'*4= 0, (»ia)'2''»3&>4 = 0, tBiWoW's W4 = 0, 

a)'i'<u'2''»3'"4 = 0; a)V'i'2™V<*'4 = Oj &)i&>'2'&)V&>4 = 0, ft) j &) 2 &) 3 '^4 = ". 

Now suppose that X is a root of &)i, then the equations remaining to 
be satisfied are 

tt)'l''*'2'"3&'4 = 0. Cl)Vft>Vw3ft>4 = 0, (»'i''B2'b'3 Q>4 = 0, ft)'i ft) 2 «» 3 '^4 = 0. 

Since tui and aj'i cannot both be zero together, X cannot make &)'i or 'cdj 
zero ; and because X is an extreme to the roots of w^, m^, CO4, X cannot make 
CO '2 or '0)2 or «3 or '&>3 or w ^ or '0)4 zero, so that in fact when « = X none of the 
singly accented quantities m can be zero. As regards the doubly accented 
quantities tu, the same thing cannot be affirmed, because if any fl contains 
only one element the corresponding value of &> with a double accent vanishes 
spontaneously. Again, any of the unaccented quantities tu may vanish, 
because we may suppose any of these to have an extreme root X. Conse- 
quently the first, second and fourth of the equations remaining to be satisfied, 
might be satisfied on making the necessary suppositions as to the form of the 
quantities to and the values of the extreme roots ; but the third remaining 
equation «u'/<"2'»V«y4 = 0, in which only singly accented quantities co occur, 
remains incapable of being satisfied on any supposition whatever. And the 
same thing would be true if we suppose X to be a root of any other « instead 
of &)i. Hence X cannot make o) = when 6 = 4. 

In like manner, if e be any even number 2e, there will be an equation 

w\' (O.^w's' (OiO) s' Wg . . . ft)'2£_l ft>2E = 0, 

to be satisfied by that value (if it exist) of x which, besides being an extreme 
(on either side) of the roots of a^, a^... m^^ arranged in order of magnitude, 
also makes to = 0. But as such equation cannot be satisfied, neither extreme 
root of the roots of o)i, oi^... w^^ can be a root of co, as was to be proved. 
Consequently, unless ^x is so assumed that the number of changes of sign 

in the coefficients of x in the quotients resulting from ^ expanded as an 

jx 



540 On a Theory of the Syzygetic Relations [57 

improper continued fraction is even (for if the changes from sequence to 
sequence are odd the number of sequences themselves is even), the method 
of limitation in the text cannot give the means of drawing either limit 
indefinitely near to one or the other extreme roots oifx. 

Art. {fji). It now remains to prove the converse, and to show, first, that 
when the number of changes is even, that is, the number of sequences odd, 
this coincidence can always be effected ; and secondly, that it is always 
possible when fx has one or- more real roots, so to assume j)X that the 
number of sequences shall be odd. 

The first part of the proposition is easily proved. Thus suppose e = 3, 
so that 

to = ft)i(B2«*3 ~ (io\'(o„(Os — (Oito^'cos + co'i'a>\ a>,. 

If we suppose \, either extreme of the scale formed by wiiting in order 
of magnitude the roots of a>i, ca^, a>sy to be a root common to (o-^ and to (03, and 
if 'co'a = 0, which last equation may be satisfied by supposing the type O2 
to consist of a single element, the separate equations 

<Bi G)2 f^3 = 0, co\'a}2a)s = 0, Ml a' 2 '0)3 = 0, a>\ 'to', '0)3 = 

will all be satisfied ; and so in general it may be shown without difficulty 
that if e = 26 + 1, and if X be a root common to 

6)1 = 0, 6)3 = 0, (D5 = . . . ftJoe+l = 0) 

and if u>2, 0)4 ... &)2€ be all simple linear functions of x, so that consequently 
'co'2 = 0, 'w'4 = . . . 'oj'oe = 0, each separate term in the development of a 
will vanish singly and separately, and consequently X will be a root of w : 
for since X makes gji = 0, wj = . . . ojni+i = 0, every product in the developed 
form ft), in which wj, a^.-.w^e+i do not each bear at least one accent, will 
vanish; and if we consider any product in which Wi, 0)3 ... toje+i are all 
accented, if in any two of these immediately following one after the other 
as <w2j;_i, Wai+i, an accent falls to the right of the first, and to the left of the 
second, the intervening term Gjjt will bear a double accent, and will therefore 
vanish, since Wat is supposed to be a linear function of x ; but it is impossible 
when every co is accented to prevent two accents of contiguous odd terms 
in any such product, from falling to the right of the left, and to the left 
of the right, term of the two, since the contrary would imply that all the 
accents would fall to the right, or all to the left, which, as above remarked, 
is impossible, on account of the two extreme terms being only simply 
accentable, that is, wi only to the right, and cuje+i only to the left. Hence, 
when X substituted for X makes &>i , 0)3 . . . (Uje+i all vanish, and when 0)2, cot... (02k 
are all linear functions of «, a; = X will be a root of «. 



57] of tioo Algebraical Functions. 541 

Art. {v). I believe that the remaining part of the proposition may be 
rigorously demonstrated, namely that when any of the roots oi fx are real, 
and the number of odd integers not exceeding the index of the degree of 
fx is m, and the number of imaginary pairs of roots in fx is /x, (^x may be so 

assumed that the quotients to %- expanded under the form of an improper 
jx 

continued fraction, may be made to take the form £1^, flj. ^3, ^4--. ^n+i, 
where D,^, Xlj ... Oj; are linear functions of x, and i is any number assumed 
at will, not less than fi, and of course not greater than m ; and where 
toi, Wa ... coai+i will have in common a root X, which may be made at will the 
greatest or the least root of &)ia)2£03...<»2t+i; the investigation, however, accord- 
ing to the present light which I possess on the subject, appears complicated 
and tedious, and therefore, in order that the press, which is waiting for the 
completion of these supplemental articles, may not be kept standing, must 
be adjourned to some future occasion. For the present I content myself 
with showing the truth of the law for the simple case where fx is a cubic 
function of x. 

Firstly. If %- gives rise to a single sequence of quotients D., we know, 

from the theory of intercalations, that it is necessary that all the roots of fx 
shall be real, and in order that when this is the case the quotients may form 
a single sequence fl, it is only necessary so to assume (/>«, that its roots may 
be intermediate between those offx. 

Secondly. If the roots of fx are not all real, or if they are all real, but 
do not comprise the roots of ^x intercalated between them, and if for greater 
brevity of ratiocination we stipulate that j>x shall have its leading coefficients 
of the same sign as that of the leading coefficient oifx, the leading coefficients 
of the three quotients will either bear the respective signs +H — , or the 

respective signs ^ h , or the respective signs H ; in the first and last 

of these cases there would be two sequences, and therefore, by what has been 
shown above, the method of limitation of the text could not give a limit 
coincident with a root. Let us then look to the remaining case, and inquire 
whether, and how, ^x may be assumed so that fx shall become representable 
to a constant factor pres by the cumulant [p(a? — a), —q{x — ^), r{x—a)\, 
where p, q, r are all positive, and a is a root oifx. 

Let this cumulant be called hfx. 

Nothing in point of generality will be lost if we suppose the leading 
coefficient of hfx to be — 1. We then have 

hfx = \_p{x— a), —q{x — ^), r (x — a)] 

= - pqr {x — a)- {x—b) — {p + r) (x — a) 



542 On a Theory of the Syzygetic Relations [57 

hfx 
and writing — = a;^ + Bx + C and making x= a, we find from the above 

identity that 

p+r = a'^ + Ba + G, that is, p = a- + Ba + G ~r, 

and pqr (x— ^) =x + a + B, 

hence ^ +a + B = 0, that is, j3 = - B — a, 

and par = 1, and therefore or = - = — — 75 -pz . 

^^ pd'+Ba+U — r 

Hence if (px be so assumed that the quotients to %- are p{x — a), —q{x — /3), 
r{x — a), we have 

h(f>x = [- g' (« — /3), r (« — a)] = — g'T- (« + jB + a) (a; — a) — 1 

= —qr (a;^ + £« — a" — a5) — 1 = {x- -\- Bx — a^ — aB + p]. 

Hence <^ (x) is of the form 

m {x"^ + Bx-a^-aB + {a- + aB + 0-r)} = m{x"' + Bx+C- r). 
If we call the three roots of /a;, a, h, c respectively, we have 

1 ^ 1 . 

'^ ~ r {a"" -\- Ba + G - r)~ r {{a - b) {a-c)-r\' 

and since q and r are both to be positive, we see that a must be taken the 
greatest or least of the three roots if they are all real, so that a- +Ba+G 
may be positive, which it will of course necessarily be if h and c are imaginary ; 
we must also have a" + Ba + G — r positive, so that the form of <J3X is 
TO {(x' — a?) + B{x — a) — t], t being necessarily positive, but otherwise 
arbitrary, a form containing two arbitrary constants, one of which is subject 
to satisfy a certain condition of inequality ; whereas when fx is of such 
a form as to admit, and <^x is supposed to be so assumed as to cause it to 

come to pass that the quotients to ^ form a single sequence, then the three 

coefficients in <f)X remain exempt from all conditions of equality but are 
subject to two conditions of inequality. And so in general when the degree 
of fx is X and the number of sequences 2i + 1, it is to be inferred that the 
n coefficients of ^a; will be subject to satisfy n — i—1 conditions of inequality 
and i conditions of equality. 

Art. (^). The theory of the determination of the minimum interval 
between either limit determinable by this method and the nearest root, 
or between the two limits so determinable when (f)X is so assumed that ^ 
gives rise to a deiined even number of sequences (which will include the 



57] of two Algebraical Functions. 543 

theory of the case where all the roots of fx are imaginary), must be deferred 
to an opportunity more favourable for leisurely contemplation. As regards 
the application of the theory to the very interesting case of all the roots being 
imaginary, the principal point remaining to be cleared up is the determination 
of the least value that can be assigned to the greatest, and the greatest 
value that can be assigned to the least root of the algebraical product 
XiX^Xs ... X„n, where X^, X.2... X^i are all of them real linear functions 
of w, subject to the condition that the cumulant [X^, X^, X^ ... X^ri] shall 
(to a numerical factor pres) be equal to a given function of the degree 2n 
in X incapable of changing its sign, which condition implies, as a necessary 
consequence, that the coefficients of x in each of the terms X-^,, X^ ... X^n 
must be affected with the same algebraical sign. 

Art. (o). It should be observed that in the application of the above 
method, the division of the series of quotients into distinct sequences 
governed by the signs of the coefficients of x is introduced for the purpose 
of drawing the limits closer to the roots, but is not necessary for the mere 
object of assigning limits. 

Thus, for instance, if there be two sequences so that 

[q:,q2...qi, qi+iqi+^. . . . qi+i'], 

and q'i+j, = vi% q\+2={vo+-j . . . qt+i' = \^—j , 

the greatest and least roots of x deduced from these equations will be superior 
and inferior limits respectively to the roots of fx ; from which it is clear that 
if leaving all the other equations unaltered, except those which contain 
respectively qt^ and q^i+i, we write in place of these 



fi+.={^+v,y 



the roots of the system of i + i' equations thus modified will a fortiori be 
limits to the roots of/«, but then the quantities 

1 1 1111 

/ii, a, + — ... /ii_i4 , p-\ , i'i + -, 1^2 + -... , 

form the same single series as would correspond to the two sequences 
qiq2...qiqi+i ■■■qi+i', 



544 071 a Theory of the Syzygetic Relations [67 

treated as a single sequence, and the same is obviously the case for any 
number of sequences*. 

Art. (tt). If we consider a single sequence as q^, q^... qn, and write 
5i = 0.1 (a; — Ci), q.^ = ai{x — c^ ... qn= «« {x — c„), 
where aj , Oa . . . a„ are supposed to have all the same sign, and write 

a,^ (a; - c,y = fi,', a,^ {x - c.f = U, + --] ... a„= {x - Cnf = {-— 

it seems not unlikely that the interval between the greatest and least of the 
roots of the above equations will be a minimum when the interval between 
any pair is the same for each pair, that is, when 

1 1 1 

/ia + — /^3 H 

^ jJa Ma _ __ H'n-i 

Oi a, tts "" a„ 

If we assume these equations, and write yOj = (h^, the equation for determining 
I will be 

[oi^, aal, a,^ ... an^] = 0. 

If « = 2 this equation becomes aittsp — 1 = 0. 
If «= 3, rejecting the factor |, it becomes 

ai eta aa^^ -(«! + %) = 0- 
If 71= 4 it becomes 

Oi asa^aj^* - (a^a^ + a^a^ + aiUi) p + 1 = 0. 

If n = 5, rejecting the factor ^, it becomes 

aia^asa^a,^'^ - {a^a„a^ + a^a^a^ + a^^a^a^ + a^^aiCi^ ^ + (01 + 03 + ^5) = 0, 

* It follows from this, that if gj, 32 ■■• 2n ^^ all linear functions of x, and if 

no root of Q can lie between the extreme roots of the function K, used to denote the cumulant 

[^/3n^ -slli, n/?3=. ±V3n']. 

the square roots being understood to be taken so as to make the sign of the eoefiScients of x 
all of them positive ; and from a preceding article we know that either extreme root of Q can 
be made to coincide with a corresponding extreme root of K. Hence we have an a priori 
solution of the following question, namely, "To determine the (n-l) positive quantities 
11 . . . tt _, , so as to make the greatest root of Q a minimum and its least root a maximum ; " 
for the greatest root of K will be the minimum greatest root of Q, and the least root of K the 
maximum least root of Q. Calling these respectively I and X, the two systems of values of 
/i , ;u„ ... Mn-i required will be obtained by substituting respectively I and X for x in the equations 



57] of two Algebraical Functions. 645 

and so in general, the equation in p being always of a degree measured by 

the integer nearest to and not exceeding - ; and it is easy to be seen that 

for all values of n, the second coefficient divided by the first will be an 
inferior limit to ^- (of course actually coinciding with it for the eases of 
n = 2 and n = 3). Hence we have the following valuable practical rule 
for finding a superior and inferior limit to the cumulant 

[fii {x — Ci), Kj {x — c.^ ... an {x — c„)], 

■where aj, a.^...an have the same sign, namely if C be the greatest, and 
K be the least of the quantities Ci, c^... Cn, G+ A will be a superior, and 
J?" — A an inferior limit, A being taken equal to the positive value of 

and it may be noticed that C and K are the quantities which would them- 
selves be the superior and inferior limits to the given cumulant if the series 
of terms a^, a^.-.an, instead of presenting only a sequence of continuations 
or permanencies, presented only a sequence of changes or variations of sign. 



Section V. 

On the Theory of Intercalations as applicable to tiuo functions of the same 
degree, and on the formal properties of the Bezoutiant with reference to 
the method of Invariants. 

Art. 56. Ifya; and ^x be any two given functions of *■ of the same degree 
m, we may form a system of m Bezoutics to / and (as shown in the first 
section), the coefficients of the powers of x™~'^, x^"^^ ... x^, of in which will 
compose a square matrix of m lines of m terms each, which will be symmetrical 
in respect to the diagonal which passes through the first coefficient of the 
first Bezoutic and the last coefficient of the last Bezoutic ; and we may 
construct a quadratic homogeneous function of m new variables, such that its 
determinantive matrix shall coincide with the Bezoutic square so formed. 
This quadratic form may be considered in the light of a generating function. 
All its coefficients will be formed of quantities obtained by taking any two 
coefficients in one of the given functions, and two corresponding coefficients 
in the other given function, multiplying them in cross order, and taking 
the difference : each coefficient of the generating function in question will 
consist of one or more such differences, and will thus be of two dimensions 
altogether, being linear in respect to the coefficients of/", and also linear 
in respect to the coefficients of 0. This generating function I term the 
s. 35 



646 On a Theory of the Syzygetic Relations [57 

Bezoutiant, and it may be denoted by the symbol B (/, (f>) : the determinant 
of B is of course the resultant to /, 4>, and the matrix to B is the Bezoutic 
square to /, <^. Now we have seen that the decrease in the number of 
continuations of sign in the series 1, -Bi («), B2 (cc) . . . Bm (ob) (where -Bi(a;), 
B2 (w) ... Bm (00) are the m Bezoutics to /, ^), as x changes from a to b, 
measures the number of roots of fx retained in the effective scale of inter- 
calations taken between the limits a and b. If we take the entire scale 
between + qo and — 00 the total number of effective intercalations will be 
the same, whether reckoned by the number of roots of y or of remaining ; 
for these two numbers can never differ except by a unit, since no two of either 
can ever come together ; but the number of each remaining in the effective 
scale will be m — 2i and m — 2i' respectively, i being the number of pairs 
of imaginary roots and pairs of unseparated real roots of /, and i' being the 
similar number for <f> ; so that we must have i = i'. 

Now obviously this number becomes measured by the number of con- 
tinuations of sign in the signaletic series 1, {B^,{B„} ... (5^), 'w^here in general 
(-Bj) denotes the principal coefficient in Bi (x). 

But (5i), {B2) . . . {Bm) are the successive ascending coaxal minor deter- 
minants about the axis of symmetry to the Bezoutic square ; and accordingly 
the number of continuations just spoken of, measures the number of positive 
terms in the Bezoutiant when linearly transformed, so as to contain only 
positive and negative squares, or in other words, measures the inertia of the 
Bezoutiant, the constant integer which adheres to it under all its real linear 
transformations. 

Art. 57. This inertia is the same number as, in the case of a homogeneous 
quadratic function of three variables used to express a conic referred to 
trilinear coordinates, serves to determine whether such conic belongs to the 
impossible class or to the possible class of conies, being 3 or in the former 
case, and 1 or 2 in the latter ; or as in the case of a homogeneous quadratic 
function of four variables used to denote a surface referred to quadriplanar 
or tetrahedral coordinates, serves to determine whether such surface belongs 
to the impossible class or to the class consisting of the ellipsoid and the hyper- 
boloid of two sheets (which are descriptively indistinguishable), or to the 
hyperboloid of one sheet, being or 4 in the first case, 1 or 3 in the second, 
and 2 in the third. The most symmetrical (but least expeditious) method 
of finding the inertia of any quadratic form is that which corresponds to the 
method of orthogonal transformations, and is, in fact, the usual method 
employed in geometrical treatises on lines and surfaces of the second degree. 
If we apply this method to the Bezoutiant B considered as a homogeneous 
quadratic function of the vi arbitrarily named variables 2*1, M2, M3 ••■«)» 
in order to measure its inertia, that is to say, the number of effective 



57] 



of two Algebraical Functions. 



547 



interpositions between the two systems of roots, we must construct the 

determinant 

d?B d-B d?B <PB 

du{- ' duidu^' duidu^'" dur^dun 



D(X) = 



d'B 

du^dui ' 



d-B 



du. 



7. + \ 



d^B 



d-B 



du„dus " du„du„ 



d^B 



d'B 



d-B 



d'B 



dumdui ' du^diu ' diimdus ' ' ' du. 



-„+^ 



All the roots of I) (A,) = 0, as is well known, are real ; the inertia of B, being 
measured by the number of positive roots of D(—X), will be equal to the 
number of continuations of sign in D (X) expressed as a function of X of the 
7?ith degree. 

If in f 00 and <j)x we reverse the order of the coefficients, and fx and (f>x 
so transformed become fiX and <})iX, it is obvious that the roots of fi and 
<f)i being the reciprocals of the roots of / and <f> respectively, the number 
of effective intercalations to /i and <^i must be the same as for / and <f). 
Accordingly we find that the form of the Bezoutiant to / and <^ is the same 
as that of the Bezoutiant oi/i and <^i, the sole difference (one only of names) 
being that B(;Ui, u^... Um-i, Wm) for the one becomes B(um, Um-i ■■■ u^, «i) 
for the other. The equation I)(\), which determines the inertia of B, 
remains precisely the same, as it ought to do, for either of the two systems 
/and (^ or/i and ^i. 

Art. 58. The theory in the preceding articles of this section may be 
made to embrace the case involved in Sturm's theorem ; for if 



and 



fx = a^x'" + a^x™^^ + ...+«„ 
f'x = ma,iX'"-~^ + (m — 1) Oja;™ 






fx = mfx -f'x 

= aia;™-! + ^a^x"^-^ + . . . + ma™, 

the Bezoutian secondaries, or which is the same thing, the simplified Sturmian 
residues to fx and f'x, will evidently be the same as those to fx and f'x. 
Accordingly, if we form the signaletic series 

fx, f'x, 5i, B....Bm-u 

where B^, fio-'-Sm-i are the Bezoutian secondaries to fx and f'x, the 
number of variations of sign between consecutive terms in this series, when 

35—2 



548 On a Theory of the Syzygetic Relations [57 

X is made + oo , will measure the number of pairs of imaginary roots in fx ; 
and fx and fx forming always a continuation, and the highest coefficient of 
f'x being supposed positive, we see that the terms of the rhizoristic series 
will be 1, (Si), (S.,)... (Sm_i), consisting of positive unity and the successive 
ascending coaxal determinants of the Bezoutian matrix to f'x and fx. Hence 
then the form of the Bezoutiant to f'x and fx will serve to determine the 
number of pairs of imaginary, and consequently also the number of real 
roots to fx. It should be remarked that the form of the Bezoutiant to f'x 
and fx, considered as a quadratic function of itj, u^.-.u-m-x and of the 
Coefficients in fx, will remain unaltered when for fx we write fx, for this 
will change the signs throughout of fx and fx; and consequently the 
coefficients in the Bezoutiant, which contains in every term one coefficient 
fromy"'a;, and one h:oxafx, will remain unaltered in sign. 

Art. 59. It. appears then from the preceding article, that for every 
function of x of the degree m, there exists a homogeneous quadratic function 
of (m — 1) variables, the inertia of which augmented by unity will represent 
the number of real roots in the given function. Now this inertia itself 
may be measured by the number of positive roots of a certain equation 
in X formed from the quadratic function (in fact the well-known equation 
for the secular inequalities of the planets), all whose roots will be real. 
Hence then we are led to the following remarkable statement. "An alge- 
braical equation of any degree being given, an equation whose degree is one unit 
lower Tnay be formed, all the roots of which shall be real, and of which the 
number of positive roots shall be one less than the total number of real roots 
of the given equation." 

Let us suppose fx written in its most general form, the first and last as 
well as all the intermediate coefficients being anything whatever : by reversing 
the order of the coefficients f'x will become fx and fx will become f'x ; the 
Bezoutiant tofx and f'x (which we may term the Bezoutoid tofx) will remain 
unaltered except in sign, and the equation of the (?)i — l)th degree in X formed 
from the Bezoutoid remain unchanged; consequently the equation in \ enables 
us to substitute, for the purpose of calculating the total number of real roots 
infx, in lieu of Sturm's auxiliary functions to fx, another set of functions 
which remain unaltered when the order of the coefficients is completely 
reversed, that is in effect, when we consider the number of real roots of 

/(-) in lieu of those of /(a;). And of course more generally the equation 

of the mth degree in \ formed from the Bezoutiant to any two functions 
fx and 4)X of the mth degree each in x, supplies a set of functions for 
determining the total number of effective intercalations between the roots 
of fx and (l>x, which do not alter when we consider in lieu of these the 



57] of two Algebraical Functions. 549 

roots of/f-J and (-) • This substitution of functions symmetrically formed 



in respect to the two ends of an equation for the purpose of assigning the 
total number of real roots in lieu of the unsymmetrical ones furnished 
by the ordinary method of M. Sturm, had been long felt by me to be a 
desideratum, and as an object the accomplishment of which was indispensable 
to the ulterior development of the theory, and it is certain that I did not 
in anticipation exaggerate the importance of the result to be attained. 

Art. 60. It may happen that the Bezoutiant to / and <^ (each of the 
??ith degree) may become a quadratic function of less than m independent 
variables, or the Bezoutoid to / (a function in x of the mth degree) of less 
than (m — 1) independent variables. This will take place whenever / and <p 
have roots in common, or whenever / has equal roots. The number of 
independent relations of equality between the roots of / and j>, and the 
amount of multiplicity, however distributed, among the roots of /", will 
be indicated by the number of orders thus disappearing out of the general 
form of the Bezoutiant and Bezoutoid in the respective cases*. In what 
particular mode the form of each would be affected according to the manner 
of the distribution of the equalities and the multiplicity requires a specific 
discussion, which I must reserve for some future occasion. 

Art. 61. I shall devote the remainder of this memoir to a consideration 
of the properties and affinities of Bezoutiants or Bezoutoids, regarded from 
the point of view of the Calculus of Invariants. For this purpose it will be 
more convenient hereafter to convert all the functions which we are concerned 
with into homogeneous forms, and I shall accordingly for the future use 
/ and (^ to denote functions each of x and y, which I shall write under 
the form 

/= a^x™ + maiX^''~^y + -|m (m — 1) a„_x™~"-y- + . . . + amy'"\ 
<j) = boX'"^ + mhiX'^-'^y + ^i{m-l) Kx^'-y- + . . . + 6„,,y™ 

In what follows a knowledge of the general principles of the Method of 
Invariants is presupposed, but a perusal of my two papers on the Calculus 
of Forms-f- in the Cambridge and Dublin Mathematical Journal, February and 
May, 1852, will furnish nearly all the information that is strictly necessary 
for the present purpose. The first point to be established is, that B, the 

* I have elsewhere defined how this word order, as here employed, is to be understood. 
If F, a homogeneous function of 3^, x, ...-t,,, can be expressed as a f auction of u-^, u^ ... «„_j 
(all linear functions of .Ti , x„ ... x^, F is said to be a function of n-i orders, or to have lost i of 
the orders belonging to the complete form. 

[t See pp. 284, 328, 411 above.] 



550 On a Theory of the Syzygetic Relations [57 

Bezoutiant oi fx and ^x, is a Co variant to the system/, ^; the variables 
in B being in compound relation of cogredience with the combinations of 
powers of x and y, 

That is to say, I propose to show that if /, g, h, k be any four quantities, 
taken for greater simplicity subject to the relation //t — ^A = 1, and if on 
substituting/c + ^y for x and hx + ky for y,f{x, y) becomes 

A^x'"' + mA^x^-^y + ^m (m - 1) A^x'^-^y- + A^^y™, say G {x, y), 

and {x, y) becomes 

B„x'^ + mB-^x'^-'- y + \m (m - 1) B^ai""-"- y- + B,n y'^, say T {x, y), 

and if B' (u^, u^ ... uj) be the Bezoutiant to G and T, B (u^, «„ . . . m,„) being 
that to / and <j>, then, on making u^, u^ ... u^, the same linear functions of 
iirl, u^ . . . u^' as 

(A + dy)'"", O + 91/^-"- {hx + ky) ...(>+ gy) {hx + ky)"'--, {hx + ky)"'-\ 

are respectively of 

«'""', x™^-y . . . xy™~-, 2/™"^, 

B will become identical with B'. I was led to suspect the high probability 
of the truth of this proposition concerning the invariance of the Bezoutiant 
from the following considerations : Firstly, that for the particular case 
where / and <^ are the differential derivatives in respect to x and y re- 
spectively of the same function F {x, y), the Bezoutiant of / and 0, which 
then becomes the Bezoutoid of F, determines the number of real factors 
in F, which obviously remains the same for all linear transformations of F. 
Secondly, that taking f and ^ in their most general form, the invariant to 
their Bezoutiant, that is the determinant of their Bezoutiant, is an invariant 
of/ and (f>, being in fact the resultant of these two functions; now as every 
concomitant (an invariantive form of the most general kind) to a concomitant 
is itself a concomitant to the primitive, so it appeared to me, and is I believe 
true (although awaiting strict proof), that any form satisfying certain 
necessary and tolerably obvious conditions of homogeneity and isobarism, 
a concomitant to which is also a concomitant to a given form, will be itself 
a concomitant to such form ; this principle, if admitted, would be of course 
at once conclusive as to the Bezoutiant being an invariantive concomitant 
to the functions from which it is derived. 

Art. 61*. Since the publication of the two papers above referred to on 
the Calculus of Forms, I have made the important observation that every 
species of concomitant, however complex, to a given system of functions, 
may be treated as a simple invariant of a system including the given system 



57] 



of tivo Algebraical Functions. 



551 



together with an appropriate superadded system of absolute functions ; thus 
an ordinary co variant involving only one system of variables, as u, v, w ... 
cogredient with x, y, z ... the variables of a system S, is in fact an invariant 
of the system 8 combined with the system i(,y — vtc, vz — wy, wx — uz, &c., 
u, V, lu ... being treated as constants ; so again a simple contra variant of 8 
is an invariant of 8 combined with the form ux + vy + 'wz + &c. ; so again, 
to meet the case before us, a covariant to the binary system /"and ^expressed 
as a function of Mj, ?*,••• w„i, where Ui, iu...u,n are cogredient with x™~'^, 
(d'^^y . . . 2/™~\ may be regarded as an invariant of the ternary system 
/, ^, fl, where 

VL — M,i2/'"~' — viU2y"''''x + ^m (m— 1) Usy™~^x- . . . -f (— )™~"'tfm,*'"~S 

(Ui, i('2...Um, being here to be treated as constants); and accordingly the 
differential equations which serve to define in the most general and absolute 
manner such covariant of /, (/>, or invariant to f, 4>, O, say /, will take 
the form 

d 

.._ , J_ X / /T I_ h 

'dbJ 



d , d 

' dcii " dbi 



+ 2{a.^ + b,i]+s(a._^ + h.^^^^ 



A 

-' da,., 



'dao^ 



+ ■ 



d ^ d „ d 

du„ dus dUi 



+ {m — 1) u. 



d 

™-^ du„ 



da,,i-i 



+ h 



db„ 



:.)- 



+ 3 a, 



' C^Mm-l 



dClm—! 



am-1 



+ b,r 



d 
dam _ 
d ■ 

aOm—s 



+ b„ 



d 



'dbr, 



+ ...+ni 






+ 2m„ 



-'dUr, 



+ 3zi„ 



2 7 

du„ 



da^ 

+ ... +(vi — 1)U2 



dbj 
d^^ 

duj I 



W=0. 



These equations may be proved to be satisfied when / is taken = B, the 
Bezoutiant to f, <j>, and thus B may be proved to be a covariant to f, <f}, 
but the demonstration is long and tedious. An admirable suggestion, well 
worthy of its keen-witted author, for which I am indebted to Mr Cayley, 
will enable us to prove the invariantive character of -B by a much more 
expeditious method. 

Art. 62. For gi-eater simplicity begin with considering functions of 
a single variable x ; and in order to fix the ideas, suppose m to be taken 
5, and write 

fx = ax^+bii:^ + cx^ + dx^ + ex+l, 

(J3X = ax^ + /3a^ + yx^ + Bx- + ex + \, 
and let ^ ="— -i- , ; this is of course an integral function of x and x', 



652 



On a Theory of the Syzygetic Relations 



[57 



since the numerator vanishes when x = x ; and we have by performing the 

actual operations, 

^ = {a^ — ha) x^ x'^ + (ay + ca) x^ x"-' (x + x) + {ah — da) a? x'- (a? + xx' + x"^) 

+ {ae — ea) xx' {a? + x'^x + xx'- + x'^) + {aX — la) («^ 4- a?x' + a;V^ + xx'^ + x^) 

+ (67 - c/3) x^x'^ + (bB - d^) a?x'^ (x + x) + (Se - e/3) xx' {a? + xx + «'=) 

+ (Z)X, - ZyS) («3 + a;V + xx''' + a;'3) 

+ {ch — dry) x-x'- 4- {ce — e<y) xx {x + x') + (cX — ^7) {x- + a;*' + x'^) 

+ (d!e — eS) xx' + (dX — 18) {x + x') 

+ (e\ - le) ; 
and if we arrange ^ under the form 

J.4,4 a?**'^ + ^4,3 x^x'' + Ai 2 «%'- + Aiia^x' + AioX* 



1 ^ H" -^0,0 



^4, 


' -^.^ 


^3, 


, A, 


-I2, 


, ^2, 


^1, 


. -^1, 


^0, 


, A,, 



A, 


i> -^4,0. 


A, 


1) -^3,01 


A, 


^, ^.,0, 


A, 


11 -^IjO 


A, 


I, ^0,0) 



+ ^2,4 ajV* + ^2,3 a^V' + il2,2 a;V + A^_ 
+ A^^i xx*' + A-i^^^xx'^ + ^i_2 ira;'^ + A-^^ 

+ ^„,4«'' +^0,3«^'' +A,2«'' +A 

it will readily be perceived that the matrix formed by the twenty-five 
coefficients, namely 

^4,2, 

^3,2, 

^2.2, 

^1,2, 

-4„,2, 

will be symmetrical about its dexter diagonal (that one, namely, which 
passes through ^.4^4 and ^0,0), and will be identical with the Bezoutian 
square corresponding to the system /, ^ ; in fact, using the notation 
previously employed in the first section, it becomes 
(0, 1) (0, 2) (0, 3) (0, 4) 

f(0^3)| f(0^4)| ^(0^5)| 

1(1, 2)j 1(1, 3)j 1(1, 4)j 

|(0, 5)\ 
[(0, 4)] + 

+ (1-4) 

1(1, 3)J + 

1(2, 3)) 
[(0,^5)|_ p^5)| p^5)| 

1(1, 4)1 1(2, 3)1 1(3, 4)1 
(1, 5) (2, 0) (3, 5) 



(0, 2) 



(0,3) 



(1. 5)] 
(2, 3)1 



(0, 5) 
(1, 5) 

(2, 5) 



(«) 



(0, 4) 
(0, 5) 



(3, 5) 
(4, 5), 



57] of two Algebraical Functions. 553 

{r, s) being used in general to denote the difference between the cross 
products of the coefficients of a;""^ and a^-^ in / and (j). Restoring now to 
in its general value, and taking /and (/> homogeneous functions of x and y, 
and making 

c^ _ /(«'> y) <f> {«!'' y') -/(^'. yO <i> (^. y) ^ 

xy' — x'y 
we see without difficulty that 

5>- = 2^,.,s [x^y'^-^-^x'^y'^-^-'}, 
where A^^g is the term ia the rth line and sth column of the Bezoutiant 
matrix to / and ^. This is the identification, the idea of which, as before 
observed, is due to Mr Cayley. 

Art. 63. If, now, we consider the system of functions 

f(^x, y) = a„«™ + maiX^~^y + . . . + a-m.y'^, 

<j) (x, y) = baX"^ + mbiX'"^~'^y + . . . + Smj/™', 

n (x, y) = Mm2/'""' - (m - 1) Km_i2/™'~' x±...+ (-)™-X«™~'. 
evidently f{x, y) (f) (x', y') —f{x', y') 4> {x, y) is a covariant with / and <^, and 
therefore (which is a mere truism) with the entire system/, ^, fl. So also 
is xy' — x'y, and therefore S-, the quotient of these two, is a covariant to the 
system. Hence, therefore, by virtue of a general theorem given in my 
Calculus of Forms, 

\dy ' dxj 
is a covariant to the system ; and, again, therefore, 

\dy' ' dx'j \dy ' dxJ 

is a covariant thereto. Now S^ is of (m — 1) dimensions in x, y and also of 
the same in x', y'. Consequently this latter form will contain only the 
quantities Mj, u^ ... u^-i, and the coefficients of/ and ^, so that the powers 
ot X, y; x, y' will not appear in it. 



Now ^= S S Ar,,[x''y'"'-^-^x''y'-^-''-% 

m-l m-1 

n™ , r. M <i\ (d \™-' , , , ( d y-"- d 



/dy 
^''idy) 






dy-"- 



dy') 



554 On a Theory of the Syzygetic Relations [57 



therefore 



(1.2.3... (m-l)}'^U/" dx'J \dy' dx) 



= S (Ar,rU%+i) + 222 (ArsUr+iUs+i), 
m-1 m-lm-1 

r and s being excluded in the latter sum from being made equal ; but this 
latter expression is the Bezoutiant to /, (f). Hence the Bezoutiant of /, (f> 
is an invariant to/, (p, X2, that is a co variant to the system /, </>, as was to be 
proved. The mode of obtaining the covariant ^, used in this and the pre- 
ceding article, is very remarkable. I believe that the true suggestive view 
of the process for finding it, is to consider 

f(x, y) 4> {x', y') -fix, y') 4, (x, y) 

as a concomitant capable of being expressed under the form of a function 
of 5^ and £B, 0) standing for the universal covariant xy' — xy\ ^ is then to be 
considered, not properly as a quotient, but rather as an invariant of the form 
^&), a function of w of the first degree, where ^ is treated as constant. 

Art. 64. B is not an ordinary covariant of / and </>, it belongs to that 
special and most important family of invariants to a system to which I have 
given the name of Combinants*, namely Invariants, which, besides the 
ordinary character of invariance when linear substitutions are impressed 
upon the variables, possess the same character of invariance when linear 
substitutions are impressed upon the functions themselves containing the 
variables ; combinants being, as it were, invariants to a system of functions 
in their corporate combined capacity quA system. That the Bezoutiant 
possesses this property is evident ; for if instead of /and ^ we write ^/+ i^ 
and h'f+ i'cj), any such quantity as arbs — agb,. (a^, br being coefficients in / 
and as, bs the corresponding ones in 0) becomes 

(kar + ibr) (k'as + i'bg) — (kag + ibg) (k'ar + i'br), 
that is (ki' — k'i) (a^^s - a^b^), 

so that B, the Bezoutiant, becomes increased in the ratio of (ki' — k'iy\ 
that is remains always unaltered in point of form and absolutely immutable, 
provided that ki' — k'i be taken, as we may always suppose to be the case, 
equal to 1. 

We derive immediately from this observation, the somewhat remarkable 
geometrical proposition, that the intersections with the axis of x made by 
any two curves of the family of curves u = Xf{x) + fi<j) (x), (/ and (j) being 
functions of x of the same degree) give rise to a constant number of effective 
intercalations, whatever values be given to X or /it for the two curves so 
selected. 

* For some remarks on the Classification of Combinants, see Cambridge and Dublin 
Mathematical Journal, November, 1853 [p. 411 above]. 



57] 



of two Algebraical Functions. 



555 



Art. 65. 5(mi, m, ... Mm) being a covariant of the system f and (j), and 
1*1, Mj... Mot cogredient with a;"*""', x'^~^y ...y'"^^, it follows from a general 
principle in the theory of invariants, that on making Wi , u„... Wm respectively 
equal to the quantities with which they are cogredient, B will become 
an ordinary covariant to f and (^. By this transformation B becomes a 
function of x and y of the degree 2 (m — 1) in a; and y conjointly, and linear 
in respect to the coefficients of/j and also in respect to those of ^. The 
only covariant capable of answering this description is what I am in the 
habit of calling the Jacobian (after the name of the late but ever-illustrious 
Jacobi), a term capable of application to any number of homogeneous 
functions of as many variables. In the case before us, where we have two 
functions of two variables, the Jacobian 

dx ' dx 



J if, <!>)- 



df 



d^ 
dy 



_dfd^ 
dx dy 



^d4 

dy dx '. 



We have then the interesting proposition*, that the Bezoutiant to two 
functions, when the variables in the former are replaced by the combinations 
of the variables in the latter, with which they are cogredient, becomes the 
Jacobianf. So in the case of a single function F of the degree m, the 

dW d W 

Bezoutoid, that is the Bezoutiant to ^5— , -5— , on making the (?n — 1) variables 



dx 



which it contains identical with 
dF 



identical with the Jacobian to 



dF 



^y...y™~^ respectively, becomes 



, , ^j— , that is the Hessian of F, namely 
dx dy 



d?F d'F 
dx'' ' dxdy 

d?F_ cPF 
dxdy ' dy^ 

As an example of this property of the Bezoutiant, suppose 
y = ax^ + bx-y + cxy'^ + dy'', 
<f> = aa? + ^x'y + lyxy- + hy^. 
The Bezoutiant matrix becomes 



a^ — ba, 



ay — ca, 



aB — da, 



ay — ca, 
iB — doC' 

+ 
yjby — c/: 

by - c/3. 



/aB — doL\ 
\by-C^J' 



aB — da, 
by — c$, 
cB — dy. 



* I have subsequently found that this proposition is contained under another mode of 
statement, at the end of Section 2 of the memoir of Jacobi, "De EUminatione," above referred to. 
t Por a strict proof of this proposition see Supplement to Third Section of this memoir. 



556 On a Theory of the Syzygetic Relations [57 

The Bezoutiant accordingly will be the quadratic function 

(a/3 - ha) ui' + {(aS - da.) + (by - c/3)} v^^ + (cS - dy) ui 

+ 2 {ay — ca) U1U2+2 (aB — da) MjMj + 2 (67 — c/3) u^u^, 

which on making 

Uj = w", U.2 = xy, 11^ = y-, 
becomes 

Laf + Mafiy + Nx-y- + P«:y^ + Qy\ (/3) 

where L, M, N, P, Q respectively will be the sum of the terms lying in the 
successive bands drawn parallel to the sinister diagonal of the Bezoutiant 
matrix, that is 

L = a/3 — ha, 

M=2(ay-ca), 

N = S(aB-da) + (hy-c^), 

P = 2(by- c^), 

Q = cB — dy. 
The biquadratic function in x and y, (/3), above written, will be found on 
computation to be identical in point of form with the Jacobian to /, <f), 
namely 

(Sax" + 2hxy + cy-) {/3x° + 2yxy + SBy-) — (Sax"- + 2/3xy + yy^) {ha? + ^cxy + dy% 
this latter being in fact 

^Laf + ZMoi?y + ^NxY + SPa:y^ + ^Qy^- 
The remark is not without some interest, that in fact the Bezoutiant, which 
is capable (as has been shown already) of being mechanically constructed, 
gives the best and readiest means of calculating the Jacobian ; for in summing 
the sinister bands transverse to the axis of symmetry the only numerical 
operation to be performed is that of addition of positive integers, whereas 
the direct method involves the necessity of numerical subtractions as well 
as additions, inasmuch as the same terms will be repeated with diiTerent 
signs. Thus if 

/= acc^ + bx^y + coi?y- + dx-y^ + ex]f' + ly^, 

(}) = ax^ + ^afy + yx^y'^ + Bx^y^ + exy^ + Xy^, 

using (r, s) in the ordinary sense that has been considered throughout, we 

obtain by taking the sum of the sinister bands in (a)* for the value of B 

when we write ar*, oi?y, x-y'^, xy^, y* in place of %, u^, Us, u^, u-^, 

(0, l)a;« + 2(0, 2) x''y + {'6 {0, 3) + (l, 2))a;y+{4(0, 4) + 2 (1, 3)} »Y 
+ {5(0, 5) + 3(1, 4) +(2, 3))«y + {4(l, 5) + 2(2, 4)}«y 
+ {3(2, 5) + (3, 4<)}x'-y<^+2{3, o)xf + {4<, 5)y\ 
* Vide Art. 62 [p. 552 above]. 



57] of two Algebraical Functions. 557 

The direct process requires the calculation of 

{bax^ + ^ha:^y + ^cx-y- + 2dxy'^ + ey*) {/3x* + iyx'y + SSx-y- + 4eexy^ + 5\y*) 

— {5ax* + if^x^y + o'yx-y- + 2Sxf + ey") (6*'' + 2cx''y + Mx-y- + iexy^ + oly*), 

each coefficient of which will contain the numerical factor 5 ; so that to 
reduce the Jacobian to its simplest form each coefficient will necessitate 
the employment of additions, subtractions, and a division, instead of additions 
merely, as when the Bezoutic square is employed. For instance, to find the 
coefficient of le^y from the above expression (a) we have to calculate 

i{25(0, 5)4-16(1, 4) + 9 (2, 3) + 4 (3, 2) + (4, 1)}, 

that is 

i{25(0, 5) + (16-l)(l, 4) + (9 -4) (2, 3)}, 

which is 5 (0, 5) + 3 (1, 4) + (2, 3), agreeing with what has been found above 
for the value of such coefficient, by a simple process of counting. The same 
remark will, of course, also apply to the computation of the Hessian of F 
by means of its Bezoutoid. 

Art. 66. This relation between the Bezoutiant and the Jacobian led me 
to inquire whether, as would at first sight appear probable, the Bezoutiant 
were the only lineo-linear quadratic function of m variables covariantive 
toy and </) (the word lineo-linear being used to denote the form of coefficients, 
such as those in the Bezoutiant, linear in respect of the coefficients in / 
and the coefficients of ^). If so, then there would have existed a method 
of performing the inverse process of recovering the Bezoutiant from the 
Jacobian, almost as simple as that of deriving the Jacobian from the 
Bezoutiant. On investigating the matter, however, I found that such is 
by no means the case*, but that there exists a whole family of independent 

* This might have been concluded immediately from the following observation. Let /, 
the Jacobian of /and <p, be expressed under the form 

A^x-'^-- + (1m-2) A-^x'^^-^y + i {2m - 2) {2m - 3) A2X°-"'-^if+... + A^^_2y'-'^-'^, 
then we know [p. 282 above] from the Calculus of Forms, that, D being taken to represent the 
persymmetrical Determinant 

J„, Jj, A„, , A^ 



A^, 



D = is the condition to be satisfied in order that J may be representable under the form of the 
sum of powers of (m-1) linear functions of x and y, and D itself is an invariant to J, and 
consequently an invariant and (as is obvious from its form) a combinantive invariant to/ and 0. 
Moreover, which is more immediately to the point, we know that the quadratic form Q 

J„»l^ + 2^l{«l(m-l)a2} + ^i,|{(m-l)^.,}g + 2u^^ ^"'"•^'J"'"^' j^3|+&c. + ^a,.-^^„^ 



558 On a Theory of the Syzygetic Relations [57 

lineo-linear quadratic covariants of m variables to every two homogeneous 
functions of x and y of the mth degree. I have, moreover, I believe, 
succeeded in determining the number of such lineo-linear quadratic forms 
for any value of m, of which all the rest, in whatever manner obtained, 
may be expressed as linear functions, the coefficients of the linear relations 
moreover being abstract numbers; in other words, I have succeeded in 
forming the fundamental or constituent scale of lineo-linear quadratic forms 
of m variables covariantive to / and ^; a result of too great interest, 
as exhibiting the affinities of the Bezoutiant to its cognate forms, to be 
altogether passed over in silence. Supposing the number of linearly inde- 
pendent forms of the kind to be v, then speaking a priori any of the forms 
taken at random might seem to be equally eligible to form one of the v 
included in the fundamental scale, combined with any (v — 1) others inde- 
pendent inter se, and of which the selected one is also independent. In fact, 
however, this is not so ; for it will always be more satisfactory to contemplate 
the fundamental scale of forms as generated successively or simultaneously 
by a uniform process ; and in the case before us, the process which I have 
hit upon, and which I believe is the simplest that can be employed for 
generating the fundamental scale, will be found not to include directly the 
Bezoutiant among the number. There will thus arise two subjects of 
inquiry; firstly, the mode of forming the fundamental scale, and proving 
its fundamental character; secondly, determining the numerical relations 

■tt'ill be an invariant to /, </> and fi (this last quantity n being defined as in p. [551]), and a com- 
binantive covariant to / and (p in the same sense precisely as the Bezoutiant is a covariant 
to the same, and like the Bezoutiant is lineo-linear in respect of the coefficients of / and <)>. 
If we operate with the symbol E, where E represents 

upon K any invariant of / and <p, we shall obtain EK, a quadratic function of jJj, Uj ••■"m. 
which by the rules of the Calculus of Forms we know will be a contravariant to / and ^, 
and the matrix corresponding to which must evidently be persymmetrieal. It is an interesting 
subject of inquiry, which I reserve for some future occasion, to determine the Co-bezoutiant, 
the Discriminant of which must be employed for K, so that when this discriminant is operated 
upon by E, the matrix corresponding to EE may become identical (term for term) with the 
matrix which is the inverse to the Bezoutiant matrix, which inverse, as Jacobi has so simply 
and beautifuDy demonstrated, possesses this persymmetrieal character. Vide the "De Elimina- 
tione," Section 5. The investigation of the arithmetical connexion between the Q of this note 
and the fundamental Co-bezoutiants must be also similarly reserved. I believe it to be generally 
true, and have verified the fact for the case of two cubic functions, that EQ gives a quadratic 
form such that the corresponding matrix is the inverse to the matrix of Q. The calculations 
necessary for extending the verification of this remarkable proposition for functions of x, y 
exceeding the third degree (notwithstanding that they are much abbreviated by the application 
of the rules of the calculus) still remain excessively laborious. The abbreviation alluded to 
consists in confining the verification in question to the comparison of either one of the two 
unreiterated terms at opposite corners of the matrix to EQ with the corresponding term in the 
inverse matrix of Q ; if these coincide, it is easy to prove that every other pair of corresponding 
terms in the two matrices must also coincide respectively with one another. 



57] of two Algebraical Functions. 559 

which connect that very important form, perhaps of all its kind the most 
important, with the forms comprised in the fundamental or constituent scale. 
These questions I propose to consider more fully at a future period. For the 
present I shall content myself with giving a method of forming the constituent 
scale (without, however, seeking the proof of all the forms extra to such 
assumed scale being linear functions of those comprised within it), and 
with determining the numerical relations between the forms in this scale 
and the Bezoutiant for a limited number of values of m. All the forms 
which we are seeking, besides being lineo-linear quadratics, must also be 
combinantive invariants to / and ^, remaining (as forms) unaltered for any 
linear substitutions impressed either upon the variables or upon the functions 
containing the variables. 

Art. 67. I must here premise that if there be any two forms of the 
same degree (and that degree odd) in x and y, a combinant may be formed 
from them, which will be linear in respect to each set of coefficients*. Thus 
calling the two functions 

aoa;™+' + (2w + l) (hx^y + \{^n-ir\) ^na^x'^'-^y^ + ... + a^+^y'^+'- 
a„a!='^+' + (2w + 1) a^af^y + 1 (2n + 1) ^na.x^-'-y"- + ... + a^mJ/^+S 
the lineo-linear combinant in question will be 
T = aoa2«+i - (2?i + 1) OiHai + J (2?i + 1) 2n a^cu„,^i 

, (2?i+l)(2n)(2ji-l) „ , 

which, using our customary notation, will be of the form 
(0, 2n + l)-(2« + l)(l, 2n) + ^-?^y^^(2, 2ji-l) + &c. 



(2n + l)(2.)(2n-l)...(» + 2) ^ ^ 
^ ' 1 . 2 . 3 . . . w 



As a corollary to this proposition (which, as well as the proposition itself, 
will be needed for the purposes of the ensuing determination), taking any 
function of an even degree in x, y, F(x, y), there will exist a combinant to 

-^r- and -T- , by virtue of what has been stated above, which will be 
ax ay 

* I may add here incidentally {although not wanted for our present purposes) that as a com- 
binant in which each set of coefficients enters linearly can always be formed to a system of 
functions two in number of as many variables and of any odd degree, so reciprocally can a com- 
binant in which each Bet of coefficients enters linearly be always formed to a system of functions 
each of the degree 2, of which and of the variables contained in them, the number is any odd 
integer [cf. p. 606 below]. 



560 On a Theory of the Syzygetic Relations [57 

Mr Cayley's well-known quadrinvariant to F ; namely, if 

F = a^x™ + 2maia;^-'?/ + . . . + a.ny'^'^, 
this will be 

2n(2n-l) , 1, ,„ 2?;. (2w - 1 ) ■■■ (w + 1) ^ 
at,a.n - ^na^a^n-T. -I ^ <^2a»-2 + • • • + 2 v") 1 2 ... w " ' 

The proposition itself is easily proved ; first, the expression T being 
expressed entirely in terms of quantities of the form (r, s) remains unaltered 
for linear substitutions impressed upon the forms / and ^ ; it remains then 
only to show that T satisfies the differential equations to T treated as a mere 
invariant, namely 



and 



" doi ^ da^ ^ da-i " ' " da^n+i 

From the hemihedral symmetry of T, which only changes its sign when the 
order of the coefficients in / and ^ is simultaneously reversed, it is obvious 
that one of these equations cannot be satisfied without the other being so too. 
Looking then exclusively at the first of them, we see that this is satisfied by 
virtue of the equations 



Hence then the differential equations to T being satisfied proves that it is 
an invariant, and, as above observed, its form shows upon its face that it is 
a combinant. 

Precisely in the same way it may be demonstrated, that to two functions 
each of the same even degree 2m as 

2m (2m — 1^ 

a,x^ + 2ma^x'^-^y + ^^ ' a^x^-^'y'' + . . . + a^my^"". 



57] of two Algebraical Functions. 561 

2m (2m — 1) 
and «„«='« + 2maia;™-'y H ^ ' a2X-'"---if + . . . + oiimy'™, 

there will be a quantity 

y, „ 2m(2j?4— 1) . - 

Cr = a.oaam - 2??iain„,,„_i H ^^ a2«2)ji-2 ± <»c. - 2maia,m-i + aoa2mi 

which, although not a combinant, will satisfy the differential equations 
necessary to prove it to be an ordinary invariant to the two given functions. 

Art. 68. Now let us consider the three forms, f, (j) and the subsidiary 

form O, where 

f= a^x^"' + ma^x'^~'^y + . . . + Umy™, 
^ = booc'^ + mbiX^^^^y + . . . + bmy™, 
fl = Mj2/™-i _ (m - 1) u^y-^-^x ± &c. + (-)'"-' Mm «'""', 

where it,, u^ ...Um are to be treated as constants. 

Make E^i+J=—, —^ ^si^^ + V^-] /, 

""^'■^ m (m - 1) . . . (m, - 2i) V dx ' dyj '' 

1 f d d \2'^+^ 

^^'+''^ = m(m-l)...(m-2t) [^dx^'^dy) '^' 

i being any integer such that 2i + 1 does not exceed m, and now consider 
E„i^-yf, E^i-t-i (f> as two functions of the degree 2i + 1 in ^, »; (« and y being 
regarded as constants); and by virtue of the formula in the last article, 
form Ti, the lineo-linear combinant of E^+jf and E^+i (j> J ^i will then be 
lineo-linear in respect to the coefficients in / and 0, and of the degree 
2 {m — {2i + 1)) in respect to x and y. Again, let 

1 id d\^ 

^''^'^m(m-l)...(m-2i+l)[^dx + '^dy) ^• 

E^iCl treated as a function of f and ij of the degree 2i will furnish a quadrin- 
variant Qi of the degree 2 (m — 1 — 2i) in respect of x and y, and quadratic 
in respect of the system ii^, u^ ... Um- We have thus two forms, Ti and Qi, 
each of the same even degree 2 (m - (2t + 1)} in respect of x, y. Forming 
between these the lineo-linear invariant Gi, Gi will be a function lineo-linear 
in respect of the coefficients of / and </>, and quadratic in respect of the 
system m,, Mj ... tt^. Moreover, Gi will (by the general principle of successive 
concomitance) be an invariant in respect to the system /, (j), O, and combi- 
nantive in respect to / and (p. Thus then Gi for all admissible values of 
i will belong to the family of forms to which the Bezoutiant is to be 
referred. 

It requires to be noticed, that when i is taken zero, so that Ti and Gi 
are of the degree 2 (m— 1), E^ for this case must be taken equal to Cl^, which 
s. 36 



562 On a Theory of the Syzygetic Relations [57 

evidently fulfils the required conditions of being of the degree 2 {m — 1) in 
{x, y), and quadratic in respect of the coefficients of Xi. If, now, m be even, 
we may take for 2i+ 1 successively all the odd numbers from 1 to (m — 1) 
inclusively, and there will be ^i forms Gi ; when in is odd we may take 
for 2i + 1 successively all the odd numbers from 1 to m, and the number 
of forms of Oi will be ^ (m + 1). It should be observed, that when m is odd 
and 2i + 1 = m, Ti will become identical with the lineo-linear combinant 
to /and <^, and Qj with the quadrin variant to fl ; and no power oi x ov y will 
enter into either, so that Gm will become simply T^ x Q„j. I am now able 
to enunciate the proposition, that G^, Gi ... G,n_ , vvhen m is even, aud 

'2 

Go, Gi ... Gm-i, when m is odd, form the constituent scale of forms, of which 

the Bezoutiant and all other lineo-linear quadratic functions of m variables, 
which are combinants of the system /, 0, will be numerically-linear functions. 
I propose to term the members of this scale Co-bezoutiants. 

As regards the present memoir, I shall content myself with exhibiting 
a partial verification of this law as regards the connection of the Bezoutiant 
with the G scale of Co-bezoutiants, and a complete determination of the 
numerical multipliers which express this connection for the cases comprised 
between m = 2 and m = 6 taken inclusively. It is impossible to predict 
for what ulterior purposes in the development of the Calculus of Invariants 
these numbers may or may not be required, and it seems to me desirable 
that a commencement of a table containing them should be made and placed 
on record. The remaining pages of this memoir will accordingly be devoted 
to the ascertainment of them. 

The theory of the Bezoutoid being included within that of the Bezoutiant, 
need not hereafter call for any special attention ; I may merely notice that 
the Bezoutoid to a function of the degree m will be a numerico-linear 
function of ^(m — 3) of the G's if m be odd, and J(m-4) of the G's if ni 
be even. 

It will be more convenient hereafter to denote the G's as G^, G^, G^ 
respectively, in lieu of Go, Gi, G^, &c., and to continue at the same time to 
give to the T's and Q's the same subscripts as the corresponding G's. 

Art. 69. Firstly. Suppose m=2, 

f= ax" + 2bxy + cy^, 
<p — aa^ + 2^xy + yy", 
n = Uiy — u,x. 



57] of two Algebraical Functions. 563 

Then 

EJ= {ax + hy)^ + {hx + cy) r,, 

E,(f} = (ax + /Sy) f + (/3a; + yy) v, 

Z = (ax + by) {^x + yy) - {hx + cy) {ax + ^y) 

= (a/3 - 6a) x^ + (07 - ca) xy + (67 - c^) y^, 

Qi = O' = V2/^ — '^UiU^xy + uio?, 

and therefore 

Q^ = (a^S — 6a) Mj^ + (^7 — ca) liiMo + (67 — C/S) lial 

Let us now form in the usual manner the Bezoutiant to /, ^ ; this is the 
quadratic function which corresponds to the matrix 

(2a;S - 26a), (a7 - ca) \ 

{ar/ - col), (267 - 2c/3)) 
that is 

i £ = (a/3 - 6a) u^ + (a7 - ca) ttjMj + (67 - CyS) ui = Gj or B = 2Gi. 

Secondly. Suppose m = 3. 

f=aa? + Sbx'y + Sexy" + dy\ 
<j)=ax^ + S^x^y + Syxy'^ + By\ 
il = Uiy- — ^u^yx + u^x''. 
We have then 

E^f= {ax^ + "ibxy + cy") ^ + {ha? + 2cxy + cZy^) ij, 
£■10 = {aa? + 'l^xy + yy^) f + (/3a;- + 2yxy + Sy^) t?, 
Ti = (aa;2 + 26a;y + cy'') {^aP- + 273.-2/ + hy'') - {hx" + 2cxy + cZ^/^) (aa;= + 2^xy + 72/=) 
= (a^ -ha) a* +2 {ay - ca) a^y + {3 (67 - c^) + (aS - da)} xy 

+ 2{hS- d/3) xf + (cS - dy) y, 
Qj = 112 = .(,^2j^ _ ^u^u^^x + (4m22 + 21*1^3) y^a? — ^lUiU^ya? + uiod^. 

Supplying for facility of computation the reciprocals of the binomial 
coefficients to the index 4, namely 

1 —J. 1 — i 1 
we obtain 

G^ = (ayS - 6a) itj- + 2 (a7 - ca) u-^u^ + {2 (67 - CjS) + | (aS - c?a)} Mj'' 
+ {(67 - c/3) + i (aS - cZa)} M1M3 + 2 (68 - d^) u^u^ + {cB - dy) u^. 

It will here and henceforth be more useful to employ [?■, s] to denote, not 
the difference of the cross products of the (r + l)th and (s+l)th entire 
coefficients in / and <p, but the difference of the cross products of these 

36—2 



564 On a Theory of the Syzygetic Relations [57 

coefficients divided each by its appropriate binomial coefficient. We may 
then write 

G^ = [0, 1] u,' + 2 [0, 2] u,u, + ([1, 2] + i [0, 3]) u,u, + (2 [1, 2] + 1 [0, 3]) u,' 

+ 2[1, 3]iwt3+[2, 3]m3'. 
Again, 
G, = {(aS - (^a) - 3 (67 - c/3)j {ihu, - vj) = ([0, 3] - 3 [1, 2]) ii,u, 

-([0,3]-3[l,2])ui. 
Hence 

G,-^G, = [0, 1] u,-' + 2 [0, 2] u,u, + 2 [] , 2] u,u, + ([0, 3] + [1, 2]) m,= 

+ 2[l,3]u,u, + [2,3]u,\ 
But, again, the Bezoutiant of/, (^ corresponds to the matrix 
3 [0, 1], 3 [0, 2], [0, 3], 

3 [0, 2], [0, 3] + 9 [1, 2], 3 [1, 3], 

[0, 3], 3 [1, 3], [3, 4]. 

Hence summing the sinister bands to form the cpefficients, we have 
5 = 3 [0, 1] u,' + 6 [0, 2] u,^u„ + (3 [0, 3] + 9 [1, 2]) u,' + 6 [1, 3] u,u, 

+ [2, 3] u," = 3G, - Gs. 
Thirdly. Suppose m = 4, 

y = aos* + 4<baf'y + %ca?y- + ^dxy^ + eif', 
<f} = aa^ + i^ic^y + 6<yx''y- + iBxy'^ + ey*, 

£1 = Vr^y^ — Suzy'^x + Biisya^ — UiO^. 
Then 

Esf= (ax + by) |' + 3 (bx + cy) ^"n + 3 (ca; + dy) ^rj-^ + (dx + ey) 7j\ 
therefore 

y ^ Uax + by) (Bx + ey) | _ ^ Ubx + cy) (yx + By) 
' \- (ax + ^y) (dx + ey)] \- (^x + yy) (ex + d^ 

= ([0, 3] - 3 [1, 2]) x^ + ([0, 4] - 2 [1, 3]) xy + ([1, 4] - 3 [2, 3]) f 
and 
Qs = (^hy - u^) (u^y - UiX) - (u^y - u^f 

= (M1W3 — Mo") 2/° ~ (mi"4 — M2M3) xy + (M2M4 — tif) a?. 
Hence supplying the binomial reciprocals 

1, -i 1, 

we have 

G, = ([0, 3] - 3 [1, 2]) (MxMa - ui) + \ ([0, 4] - 2 [1, 3]) (%m, - u^u,) 
+ ([1, 4]-3[2, 3])(m,M4-W3^). 



57] of two Algebraical Functions. 565 



— {aa? + S/3x'y + Syxy'' + S2/') (6«' + 3ca^3/ + ScZa;?/" + ey^) 
= [0, 1] *■« + 3 [0, 2] afiy + (3 [0, 3] + 6 [1, 2]) «*y= + ([0, 4] + 8 [1, 3]) a^f 
+ (3 [1, 4] + 6 [2, 3]) «y + 3 [2, 4] a;^/^ + [3, 4] y% 
and 

= Mi^^/" — QlliU^y^OC + (9«2^ + 6UiUs)y*x'' — (2UiUi + ISMaMs)^^^ 

+ (9^3^ + 6M2M4) 2/^^ — Qu^u^yx^ + u^a?. 
Hence, supplying the reciprocal binomial coefficients, 

we find 

Gi = [0, 1] ii{- + 3 [0, 2] ?«,Wo + (i [0, 3] + 1 [1, 2]) (9it/ + Qu^u,) 

+ (tV [0, 4] + T§T [1, 3]) (Ml 1^4 + 9M2M3) 

+ (i [1. 4] + 1 [2, 3]) (9^3' 4- 6M2M4) + 3 [2, 4] u,u, + [3, 4] u,\ 
Now the Bezoutic square, taking account of the binomial factors in / and ^, 
may be written under the form 

4 [0, 1], 6 [0, 2], 4 [0, 3], [0, 4], 

[0,4], 4 [1,4], 6 [2, 4], [3,4]. 

Hence the Bezoutiant B becomes 

4 [0, 1] Mi^ + 12 [0, 2] M1M2 + (4 [0, 3] + 24 [1, 2]) ui + 2 [0, 4] u^u^ 
+ (2 [0, 4] + 32 [1, 3]) W0M3 + 8 [1, 4] u^u, + ([1, 4] + 24 [2, 3]) <- 
+ 12 [2, 4]wsM4 + [3, 4]m/. 
And we ought to have B — cG^ + eQ^, to satisfy which equation we must 
manifestly have c = 4 ; to find e, compare the coefficients of u^, this gives 

4 [0, 3] + 24 [1, 2] = ^ [0, 3] + ^ [1, 2] + e(3 [1, 2] - [0, 3]) ; 
accordingly we ought to be able to satisfy the two equations 

im _ e = 4, ^ + 3e = 24, 
each of which accordingly we find is satisfied by the equality e = -^. 
Substituting in the equation for B above written, we thus obtain 
-B=4ffi+J^e3. 
which will be found to be identically true. 



566 On a Theory of the Syzygetic Relations [57 

Art. 70. We may now see our way to a more concise mode of obtaining 
the numerical coefficients, by which they may in fact be computed and 
verified with comparatively little labour, connecting the Bezoutiant with the 
Co-bezoutiant forms of the constituent scale. It will not fail to have been 
remarked, that throughout the preceding determinations I have presumed 
the truth of the formula, which admits of an immediate verification, that for 
all values of m and o) we have the identical equation 



+ »; T- J \c^x^ + mci x™~^ y + \fm (m — 1 ) C2«"'~y +...+m Cm-i^y'^~^ + Cmy™ r 



d^ . d 
dx 

= m (to - 1) . . . (m - o) + 1) -jZof " + (bXi|"-177 + ^co{cc- 1) Z2^"-V+ • • • +L^V'' \ , 

where 

i(i=Coa;'""'"+ (m - a) CiX^~''~'^y + (to — to) ^ c^x^-'-'-^y^ + . . . + Cm-o.J''" 

i/i = Cia;™~" + (to - a))c^'^~''~^y +{m—o>) "~ Cja;'"-"-^ + . . . + Cm-^+iy'" 



i„=c„«™-"+ (m- a) c^+i*™"'^'^ + {m — a>) ^ CiX^-"^y- + . . . + 0^2/' 

Let us now proceed to determine by an abridged method the linear relations 
corresponding to the cases of to = 5, m = 6, and first for m = 5. 

Let 

/= ax^ + 5baf^y + lOca^y^ + lOdx^ + 5exy^ + hy^, 

(j) = aa^ + b^odhj + lO'ya^y'^ + lQha?y^ + 5exy* + riy^, 

O = Ur^ — A^u^'x + Qib^y^x- — ^Uiya? + u^x'^. 

In forming G^,, G3, Oi, let us confine our attention to the terms ui', u^tis, iiaih- 

A comparison of the coefficients of these with those in the Bezoutiant {B) 
will be sufficient for assigning the three numerical quantities which connect 
B with G'l, G3, G5. I omit U1U2, because G^ is the only one of the G's for 
any value of to which contains Ui" or MjMs, and in Gi the terms containing 
Ui" and ttiMj are 

[0, 1] Ml" + (to - 1) [0, 2] M,M2, 

and the corresponding part of the Bezoutiant is 

to [0, 1] Ui' + TO (to - 1) [0, 2] MiWj ; 
so that if we write 

B = CiGi + CsGs + CjGs + &c., 

the two terms u^- and Ut^u^ will only enable us to form one equation with the 
c's, namely, Ci = m. Again, instead of considering the entire coefficients 



57] of tioo Algebraical Functions. 567 

of z/jMa and u-^Ui, it will be sufficient to take a single argument of either 
of these coefficients (in the forms to be compared), as for instance [0, 3] and 
[1, 3]. Then Cj being known, Cj, Cj will be determined; but for the purposes 
of verification I shall furthermore compute the whole of the coefficient of u^u^. 

Accordingly, calculating the G system in reverse order, we have 

G^ = {[0. 5] - 5 [1, 4] + 10 [2, 3]} [u^u, - 4zt,M, + ^i,,i] 

= {[0, 5] - 5 [1, 4] + 10 [2, 3]} u,u, + ..., 

E^f= {aa? + 2bxij + cy^) ^= + 3 (bx'' + 2cxy + dy"") p»? 

+ 3 {ex' + 2dxy + ey-) ^ij- + {da? + lexy +/y^) rf 
^s0 = &c. &c.; 

therefore 

Ts = {{ax- + 2hxy + cy") {hx" + 2exy+ rjy-) - {ax^ + 2/3xy + yy") {dx- + 2exy+hy'')] 

- 3 {(6«2 -I- 2cxy + ^2/2) {r^x" + 'ihxy + ey-) - {^x- + i^^xy + Sy") {cx^ + 2dxy + ey"")] 

= ([0, 3] - 3 [1, 2]) «^ + (2 [0, 4] + . ..) x'y + {[0, 5] + [1, 4] - 8 [2, 3]} x'y'' + &c. 

The number — 8 results from the calculation 1 — 3 (4 - 1) = — 8. 

Again, 

E„rL = {u^y^ — 2u2yx + tisof) p — 2 {u.^y^ — ^u^yx + UiOf) ^n 

+ {u^y^ — 2Uiyx + u^x'^) rf, 
therefore 

Q,i = (^2/^ — 2u^yx + Ms*-) (Ws^^ — 2Uiyx + u^af) — {u^y'^ — iu^yx + u^a?■)'' 

= ihUsy* — 2uiUiy^x + UjU^y-x^ + &c., 

all the terms and parts of terms unexpressed being free of Mj, and therefore 
not necessary for our purpose. Hence supplying the reciprocal factors 

1, -h h -. 
we have 

G, = [0, 3] u^u, + ([0, 4] +) u,u, + ^ ([0, 5] + [1, 4] + [2, 3]} u,u, + &c. 

Again, expressing ^j/and E^cf) in the usual way, we obtain 

Ti = {aog* + ^bx'y + Qcx^y- + idxy^ + ey^) {0af + ^r^a^y + 6Bx''y^ + 'iexy^ + ijy*) 

- {oue^ + 4<^x^y + 6ya?y^ + 4^Sxy^ + ef) {bx^ + 4,cx^y + Qdx^y'^ + 4<exy^ + hy^) 

= [0, 1] «« + 4 [0, 2] x'^y + (6 [0, 3] +) x'f- + (4 [0, 4] +) x'y^ 

+ ([0, 5] + 15 [1, 4] + 20 [2, 3]) x'y' + &c. 

(where it may be observed that the numbers 15 and 20 in the coefficient of 
a^y^ arise from the quantities 4^^ — 1, 6^— 4^). 



568 On a Theory of the Syzygetic Relations [57 

Again, 

Qj = X12 _ y^ir^ j^ SuiU^x'y + l^u^u^x^y'^ — SuiU^x^y^ + 2uiU^a^y* + &c. 
Hence supplying the multipliers 

,-11-1 1 , 

^' "8"' 28- 56"' +70- ^*'- 
we have 

G, = [0, 1] %^ + 4 [0, 2] u,u, + Jf [0, 3] n,u, + \ [0, 4] u.,u^ 

+ h ([0. 5] + 15 [1, 4] + 20 [2, 3]) u^u,. 
Again, the Bezoutiant 

5 = 5 [0, 1] V + 2 . 10 [0, 2] u^u^ + 2 . 10 [0, 3] WiWg 

+ 2 . 5 [0, 4] itiM4 + 2 [0, 5] M1M5 + &c. 

Accordingly, if we write 5 = CiGi + CsG^s + CsG^j, we have, as above remarked, 
Ci = 5 ; and to determine C3, Cj, we have, by comparing the coefficients of 
M1M3, u^Ui in 5, (ri, (ts, 6^5, 

20 = ^ + 0, 

10 = ^4- C3. 

These two equations, then, as it turns out, are not independent, but are 
satisfied simultaneously by 

^ _ 50 

Finally, equating the coefficients of the several arguments in M1M5, we have 
= 5xJ5+-^x^ + C5 from the argument [0, 5], 
= 5 X II + ^ X ^ + Scj from the argument [1, 4], 
= 5x|A + ^xf + IOC5 from the argument [2, 3]. 

The first of which equations gives 

the second gives 
and the third gives 

n 20 12 2 

C5 - -JT + 7 — t- 

We have thus abundantly verified the accuracy of the calculation, and there 
results the relation 

Lastly, let m = 6, 

/= a*'" + 'obaf'y + 15ca^y^ + 'i.Qdx^y^ + Iheoi^y* + Qhxy^ + ly'^, 
<f) = ax" + 6j3x^y + 15'yx'^y^ + 20S!e'y^ + 15ex-y* + Qrjxy^ + \y^, 
fl = iir^y^ — bu„y*x + lOii^y^x' — lOuty-af + Busyx* — u^af. 



57] , of two Algebraical Functions. 569 

I shall here confine myself to the determination of a single argument in 
each of the terms u-^, UiU^, UiU^, M1M4, UiU^, u,tig; this will be ample for the 
purpose of verification, as the equation to be assigned is of the form 

The arguments which I select as the most simple, will be those expressed by 
the symbols (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6) respectively; then we have 

Ti, = (ax + by) (rjx + \y) + &c. — (hx + ly) {ax + ^y) 
= ([0, 5] + ...)^ + ([0, 6] + ...)«^2/ + (.•.)2/^ 

Qs = {u-iy — ^«2«) (wsj/ — M6«) + &c. 

= (mi «5 + . . .) 2/^ - {uiUs + ...)yx-ir{...)x\ 
Hence supplying the binomial reciprocals 
J-, 2> ■■■' 

G, = {[Q, 5] + ...)«! Ms + i([0, 6]+...)i<iM6 + &c. 

Again, 

T3 = {aa? + ...) {^a? + Zex-'y + 37)xy- + \y') + &c. 

- (dafi + 2ex'y + Shxy- + ly^) (aa^ + . . .) 

= ([0, 3]+...)^'' + (3[0, 4] + ...)«'2/ + (3[0, 5]+...)«'2/' 

+ ([0, 6]+.,.)«'2/'+&c. 

Qs = {u-^y^ + &c.) {u^y^ + Zuiy'^ + Zu^a? — u^x^) — &c. 

= {uiUs+ ...)y^— (3mi«4+ ...)y^x + {ZuiU^+ ...) 2/V — (mjMs + ...)y^x?+ &c., 

and the reciprocal binomial multipliers will be 

-1 +1 -1 
^' -6-' 15' 20 ■'^°- 
Hence 

G^ = [0, 3] 2«iM3 + 1 [0, 4] u^Ui + f [0, 5] M1M5 + ^V [0, 6] ttiM^ &c. &c. 

Finally, 

Ti = (a«« + &c.) (/3a;= + o^*'"?/ + lOha^y^ + lOea;^?/^ + Si^ajj/^ + \y^) - &c. 

= ([0, 1] + ...)«'» + 5 ([0, 2] + ...)«''y + (10[0, 3] + ...)a^2/' 

+ (10[0, 4]+...)«y + (o[0, 5] + ...)«y + ([0, 6] + ...)«y + &c. 

Qj = 122 = ,^^2yio ^ (io,,^M,. + . . .) y9^ + (20tiiM3 + . . .) 2/Sa;^ + (20^11^4 + . . .) 2/V 

+ (IOM1M5 + . . .) y^ar* + (2ttiMs + . . .) 2/V + &c. ; 



570 On a Theory of the Syzygetic Relations [57 

and supplying the numerical series 

1 -1 1 ^ JL ni ^ 

10' 45' 120' 210' 252' ' 
we have 

G, = [0, IJ u,^ + 5 [0, 2] WiM, + -4^0 [0, 3] w,M3 + S [0, 4] u, u, 

+ ¥T [0. 5] u^u, + ^ [0, 6] UiMs + &c. 
Again, the Bezoutiant 

= 6 [0, 1] u,^ + 30 [0, 2] u,u, + 40 [0, 3] u,u, + 30 [0, 4] u,u, 

+ 12 [0, 5] u^u, + 2 [0, 6] ihu, + &c. &c. = B. 
Hence making 

B = c^G-, + c^Gs + c,G,, 
from itj2 and WjZta we obtain respectively 

Ci = 6, 
5ci = 30 ; 
hence from u^u^ and M1M4 we obtain respectively 
140+ c, = 401 



^+|C3 = S0J °''=-^ 
hence from MiJtj and u-^u^ we obtain respectively 

6 X ^T + ¥ f + C5= 12, that is 05= 12 - 8 -JjP- = ^, 

6xTk + ^^+ic5 = 2, thatis|c, = 2-f-2V = f; 
hence 

^ — 18 

and the equation sought for is 

B=QG, + ii-G,+^G,. 

Art. 71. The following table exhibits the relations between the 
Bezoutiant and the correspondent system of Co-bezoutiants for all values 
of m between 1 and 6 under a synoptical form. 

m=l, -B = Gx, 

m = 2, B=2G„ 

m = 3, 5 = 36^1-6^3, 

m = 4, B = iG^ + ^G„ 

m=5, B = 5G^ + Ap-G, + ^G„ 

m = 6, B=GG, + ^G, + i^G,. 



57] of two Algebraical Functions. 571 

These series could if wanted be easily extended, and the calculation of the 
coefficients reduced to a mere mechanical procedure. 

If we suppose m to be 2i or 2i— 1, we have the equation 

5 = Ci Gi + Cs (?3 + . . . + C2{_i (T2i_i, 

and it appears from the foregoing instances that the comparison of the 
coefficients, either of m,^, or of u-^u^ on the two sides of the equation, 
will serve to give Cj (m being known), Cg may be found by a comparison 
of the coefficients either of u^u^, or of u^Ui, and so on for c^.-.c^i-^; 
all the coefficients in the equation for B above given, thus admitting of being 
found separately and successively and in two modes, so that there is a check 
at each step upon the correctness of the computations : the only exception 
to this last remark is (when m is odd) for the last coefficient of which the 
above condensed method affords only a single determination. I need hardly 
add the remark, that in substituting a;™~S x™-'^y, . . . xy'^-^, y"^^ in place of 
tfi, 1*2 ... Um-i, Urn, respectively, all the G's become (to a numerical factor jjres) 
identical with one another and with the Jacobian to the system (/, <^). 

Art. 72. The foregoing theory took its origin (as will have been readily 
imagined) in meditations growing out of the celebrated theorem of M. Sturm. 
There appear to be several directions in which a development or extension 
of the subject matter of that theorem may be sought for. Thus a theory may 
be constructed relative to a single function of one or more variables, viewed 
in all cases as representing a geometrical locus. In the limiting case, when 
this locus becomes a system of points in a right line, we have the theorem 
of Sturm ; generally the theory will be that of contours. Or, again, a theory 
may be formed in which the number of functions is always kept equal to that 
of the variables. We have then a theory of discrete points corresponding 
to roots, the number of real ones of which comprised within given limits 
it is the object of such theory to determine. M. Hermite, in a memoir 
recently presented to the French Institute, appears to have made a valuable 
addition to the Sturmian theory extended in this direction, to which the 
beautiful researches of M. Cauchy and the joint labours of MM. Liouville and 
Sturm, with reference to the disposition of the imaginary roots of equations 
appear to have led the way. Finally, the number of variables may be supposed 
to be arbitrarily increased, but made always inferior by a unit to the number 
of the functions in which they are contained, or which comes to the same 
thing, we may construct the theory of a system of homogeneous functions 
equal in number to the variables in them, which in its simplest case becomes 
the theory of Intercalations which has been here partially considered, and 
which (as has been shown) embraces (not as a particular case, but as an 
implied consequence and easily extricated result) the theorem of M. Sturm. 



572 On a Theory of the Syzygetic Relations [57 



General and Concluding Supplement. 

Art. (X)- The expressions given in Art. (n) [p. 507 above] for the partial 

quotients of the continued fraction represented by ^ , are restricted to the 

supposition of all these partial quotients (except the first) being linear in x ; 
when the fii*st partial quotient is linear the formula (B) of that article continues 
applicable on replacing (Bihg) by 1. I was forcibly struck by the peculiarity 
of these formulas not ceasing to be true in consequence of the first partial 
quotient being supposed non-linear ; and reflecting upon this, I was soon led 
to perceive that all the partial quotients might be supposed to be arbitrary 
integral functions of x, and the formulae would still continue to apply to 
any such of them as might happen to be linear, although, as it were, imbedded 
among a group of other non-linear partial quotients. From this it was but 
an easy step to perceive that the formulse (A) and (B) must admit of extension 
to the representation of partial quotients of any form, and that the dimorphism 
of the representation of the linear partial quotients could only be a consequence 
of the equation in integers u + v = l having two solutions w = 0, v=l and 
u=l, V = 0. I now proceed to enunciate the very remarkable general 
theorem (or as it may perhaps not inappropriately be termed Algebraical 
Porism), by virtue of which any partial quotient of a given degree in x 
belonging to an infinite continued fraction, all of whose partial quotients are 
algebraical functions of x, may be expressed to a constant factor pres, by 
means of the numerator and denominator (or if we please either one of these) 
of the convergent immediately antecedent to and of the numerator and 
denominator of any convergent not antecedent to the partial quotient which 
is to be determined. 

Art. (3). Theorem. Let Qi, Q«... Qi, Qi+i ... Qn, &c. each of an arbitrary 
degree in x, be the n first partial quotients of an algebraical continued 
fi-action ; let Q^+j be the partial quotient to be determined and of the given 
degree Ui+i', let 

J^ 1 1_ 1^ _ <j>i (x) 



and 



J 1 1 1^ J_ J^ ^ ^i^) . 

l-Q^Q^Qi-Qi+."' Qn F{xy 



let u and v be any couple of integers of the (»i+i 4- 1 couples which satisfy the 
equation v+u = Wj+i ; then, as usual, denoting the product of the differences 
of each of one set of terms from each of another set, by writing the former 
under the latter, and calling ri^, t], ...tj^ the fj, roots of ^(x), and h^, h^ ... A,„ 



57] 



of tivo Algebraical Functions. 



573 



the m roots of F (a-), (<& and F being supposed respectively of fi and m 
dimensions in x), and forming the disjunctive equations 

ti, U, ts ...t,n =1,2, 3 ... VI, 

we have the following equation, wherein (p and y are written for (pi and /J-, 



hr^^l, hu+2--- K,_ 


X 




Vex, Vei •■■Ve^ 
Ve^^.v '79^+2 •■• ■^e^. 


X 


hi, hi ' ■■■ hu 

A„4.,, hu+2---Jjt„,_ 



X K* - Vei) (^ - '7^2) ■■■(x- Ve,)] {(« - ^t,) (« - /i«2) •■•(«- /i(u)} [ , 



and moreover the different values of K^^^ depending upon the different modes 
of breaking up coi+i into two parts it and v are all (to a numerical factor pres) 
equal to one another. Thus then the theorem pointed at in Art. (p) is 
discovered, and the way laid open (by an unexpected channel) for a complete 
discussion of the theory of the singular cases which may occur in the 
expansion of any rational algebraical fraction under the form of a continued 
fraction. 



Art. (J). In the above expression, if we suppose Wi+i = 1, we have m = 1 
and V = 0, or u = and v = l, and remembering that 

P ]=<DAand [j , , 1 = i^'/, 

h^'K-'-KJ IVe^: Ves---Ve,.j, 

Qi+i becomes by virtue of the general formula representable under either 
of the equivalent forms 

K^^ao \{<j>ney §^^ {X - ve)] and ^,.„ 2* |(/A,)= || {x - h) 

ifo,! and K-i^a being either equal, or differing only in the sign, agreeably to 
the formulae (A) and (B) [p. 508 above]. 

Art. (^). It may be worth while to notice, that, although (of course) 
these formulas and the general formula of Art. (3), when supposed converted 
into functions of x and of the coefficients of F and of <3E> by the reduction, 
integration and summation of the symmetrical functions of the roots which 
enter into them remain universally valid, and subject to no cases of exception, 



574 On a Theory of the Syzygetic Relations [57 

yet antecedently to these processes being performed the formulse as they 
stand may become illusory when any relations of equality exist between the 
roots of <5 inter se, or between the roots of F inter se. Thus in the case before 
us, if <5 have equal roots the formula commencing with iTo.i is illusory, and 
if F have equal roots the other of the two formulse becomes illusory. 

Let us take the second of these and suppose that F{x) has 

^'i roots Ci, kz roots Cj ... kp roots Cp, 

we may pass to the actual case from any case where the roots are infinitesi- 
mally near to the actual roots of F{x), and all infinitesimally different from 
one another. Moreover the choice of the infinitesimal variations being 
arbitrary, let the fci roots Ci be replaced by a group of roots 

Ci +8, Ci + S/3i , Ci + 5/91= . . . Ci + S/o^-S 

where p^ is a prime root of the equation p^^ = 0, and S is an infinitesimal ' 
quantity, and suppose each of the other groups to be varied in an analogous 
manner. Then it may easily be shown from this that the second of the 
formulse in question will become 

^ (jX'KA)=(*cO(^-c,)} 
and similarly, the twin formula becomes 



, U^enFye){«^-^e)} 

(I.) *- 

Cori-esponding modifications will admit of being made by aid of a like 
method in the general formulse of Art. (3) upon a similar supposition as to 
equalities springing up between the roots of fao per se and of ^ {x) per se, 
or between the roots oifx and <^a; inter se. 

* For in general if p is a prime root of the equation p'" = 1, and if fx have w roots all equal 
to c and ^x is any other function of x and if 5 is an infinitesimal quantity, then rejecting all 
powers of 5 higher than the (w — l)th degree, 

f {c + d)'^ f'(c + pd)^ f'{c + p''dr -^ f {c + p^-'d) 
{tj/{c + S) + p4>{c + pb)+p''iy{c + p'S) + ...+p"-^i'{c + p'-'-^S)} 



(i)v-- 



67] 



of two Algebraical Functions. 



575 



Art. (n). If in Art. (3) we take i = 0, the formula for Qj+i will become 

h^ ... ht„ 
Ve.+i •••Veu 



Q^=K„, 






■1 X P- 



■■■VeJ\^U>'h' K 



iVe^+i > ■'/9„+2 • • • Ve^iJ L"'t<'+v "tu+2 



u and V being any two integers whose sum is toi, which is identical (as it 
ought to be) with the expression virtually contained in the formulae of 
Section II. for the syzygetic multiplier of <I> («) in the syzygetic equation 
connecting Fx and <!>« with their first residue when <!>« is supposed to be o>i 
dimensions in x lower than i^« identical, videlicet, in other words, with the 

integer part of the algebraical fraction ^^< . 

Ai-t. (1). When * {x) = F' (x), 
^rl^^'^y''^'"^?"'^! becomes identical with (-)J K+i-^' "i+i ^{h,, h,...h^J, 

\Jh+<^i+i, h+<^i+i ■ ■ • hm J 

and we may consequently (using an extreme term in the forms in the 
polymorphic scale of forms representing Qi+i), write 

Qi+, = {-)i '"m-i "« K,,^,, ^K{K h... h^J (fAf (/Ay . . . 

(fiK+^y {x - k) {x-h^)...(x- h^^^). 

Art. (T). The following observations will serve to complete the theory of 
the singular cases in the expansion of an algebraical continued fraction. 

Preserving the notation of Art. (3), let 

ffi = m — (cDi + (02+ ... + «i_i + 1). 

Then (calling the roots of Fx, h^, h^ ...hm) the (i)th simplified residue to 

=-, in accordance with the general formulse for the residues in the second 

section (for greater simplicity selecting an extreme term of the polymorphic 
scale), will be represented by 

S ^, ^ ^ ^ (a; - Ai) (« - ^=) (« - ^s) ••• («-ft<ri), 






which will be of the form Zi^'r"."*"' + &c., all the terms containing higher 
powers of x vanishing by the coefficients becoming zero. If in the above 
expression we should use o-/ in lieu of o-j, where a-;' is o-j diminished by any 
integer inferior to «<, we should get other forms of the same residue, but 



576 On a Theory of the Syzijgetic Relations [57 

these will all be of higher dimensions in the roots or coefficients than 
the one just given, and in fact the forms thus obtained corresponding to the 
values o-i, o-i - 1, o-; — 2 . . . o-j — «; + 1 substituted for o-f in succession, would, 
by aid of the relations of condition between the coefficients of <^x and Fx 
implied in the value of coi, admit of being exhibited as a scale in which each 
form would be an exact algebraical product of the form which precedes it, 
multiplied by a function of the coefficients, and did space permit thereof 
it would be perfectly easy to give the forms of these multiplicators. But 
I pass on to the representation of what is more material, namely, the form 
of the complete residue in the case supposed, merely observing (as an 
obiter dictum) that the existence of each singular partial quotient (meaning 
thereby a quotient non-linear in x) only affects the form of the single 
simplified residue in immediate connexion with itself, and not at all the form 
of the other residues antecedent or subsequent to that one. 

Art. (n)- Let the ith simplified residue be called Ri and the correspond- 
ing complete residue [jRj], then applying a method similar to the method 
given in Section I., we shall find that 

Li representing the leading coefficient in the iih simplified residue, and the 
sign of interrogation (?) denoting some function of Wi, &)2 ... cdj (possibly a 
constant) remaining to be determined. And reverting to Art. (1), the 
quantity that would be called K^^ ^ according to the notation employed in 
the formulae expressing Qi+i in that article, will (abstraction being made of 
the algebraical sign and using for greater brevity (t), (i — 1), &c. to express 
1 + Wj, 1 +a)i_i, &c.) come to be represented by 

zi" 4i'-^'i:iv^'&c.' 

a similar convention being supposed to be made respecting the numerator 
and denominator of each convergent as was made respecting them in the 
particular case treated of in Art. (/), page [502]. 

Art. (^). I will merely add a very few words in generalization of the 
method of limiting the roots of /« given in the Supplement to the fourth 
Section [p. 528 above]. As an inferior limit to fx is identical with a 
superior limit to /(— x), we may confine our attention to superior limits 
alone. Suppose then that 

^__J 1^ J^ 1 1 J_ 1 1 1 

>~Qi-Q.-'"Qi-Q/-Q;-"'QV'(Q)i-(Q)2-""(Q)w' 



57] ! of two Algebraical Functions. 577 

where the partial quotients Q are each of any arbitrary degree in x, and have 
all one algebraical sign in the coefficients of the highest powers of x from Q^ 
to Qi, ahd all the same sign (contrary to the former), in the coefficients of 
the higJiest powers of x from Q/ to Q'j-, and so on alternately, then firstly 
a superior limit to the superior limits of the cumulants [Qi, Q2 ■■■ Qi\, 
[Qi, Qi ■■■ Qi'], ■■■ [(Q)i. {Q% ■■■ (Q)(i)] ■wi'l ^^ ^ superior limit to fx, so that it 
remains only to give a rule for finding a superior limit to a curaulant 
[Qi. Q-i, Q3 •■• Qi]: which, secondly, is to be found by making 

Q,-M, = 0, Q,-i/, = 0, Q,-M, = O...Qi-Mi = 0, 

1 11 

where M, = /j,„ M^ = ijl, + — , M^^ h^-{- — ... Mi = , 

/Xi, /^a ... Mt-i being any quantities entirely independent and arbitrary except 
in regard to their being all of the same sign as the leading coefficients in the 
elements Qi, Q2 ... Qi- 

"We may then find L^, L^... Li any superior limits to the roots of x in 
these i equations respectively ; L, the greatest of these, will be a superior 
limit to the proposed cumulant [QuQ^... Qi] ; and it may be observed that 
M^, M^...Mi are the general values which satisfy the equation 

1 ]_ JL^A 

^' M,-M,--Mi "' 
subject to the condition that for all values of e 

11 1 J^ 

shall have a given invariable sign. The first part of the process, as just 
shown, consists in separating the t3rpe of the total cumulant which represents 
fx into partial types, the point for each fracture of the total type being 
marked by a change of sign in the elements of the tjrpe for the value 

a; = + 00 ; it is easily seen therefore from this, that if y- is the generatrix 

of the cumulant in question, the number of such fractures (that is, the 
number one less than the number of partial cumulants) will be the number 
of changes of algebraical sign in the signaletic series, consisting of the 
leading coefficients in Fx and in each of the odd-placed complete residues 
respectively, together with the number of changes of sign in the signaletic 
series, consisting of the leading coefficients in <!>« and in each of the even- 
placed complete residues respectively. 

The syzygetic theory of two algebraical functions, and the allied theory 
of algebraical continued fractions with their principal applications, may, 
I think, now be said to be completely made out, as well for the singular 
cases as for the general hypothesis. 

s. 37 



578 On a Theory of the Syzygetic Relations [57 

Art. (*). I will conclude with observing that the theory within developed 
gives the means of transforming (explicitly and without the aid of sym- 
metrical functions) into an algebraical continued fraction, any given sum of 
algebraical fractions of the form 

Ci C2 C3 Cfi 

X — hi X — h^ X — h^ " X — hn' 
where each c and h is supposed known. For let the above sum be called 
p— , then a hg, Ce be used to denote any pair of corresponding terms of the 

h series and the c series, we have ^j^tt- = Cb, as is well known and easily 

£ hg 

proved. Again, ii DiX represent the simplified denominator of the ith con- 

^x 

vergent to the continued fraction equal to -p- which is to be found, say 

11 1 



we have [p. 476 above] 
DiX = S j-T — ' , ^ " " , ' {x — hi) (x- hi) ...(x — hi) 

i'izI>^Qii,h2...hi)^h,^\...'i>hi. ,,. -, , ,. 
= S(-) 2 ^-^ F\F%...F% (a^-hi)(x-K)...{x-h) 

■ i-l 

= (-)' 2 S {C1C2 . . . Ci ^{hi,h,... hi) (x - hi) (x-h„)...{x- h)}. 
Therefore 

(DihiY = (S (c,c3 . . . Ci+0 KiKK... hi+i) {K - K) {hi - h,) ...(hi- hi+i)Y 

= {2 (C2C3 . . . Ci+i) ^'^(hi,K... hi+i) f^ Qi^, hs... hi+i)Y ; 

and the simplified (i + l)th quotient, that is, the value of Ai+iX+ Sj+i, when 
divested of the allotrious factor, has been proved [cf p. 508 above] to be equal to 

tiDihiY^ix-hi); 

it is therefore now known as a rational and integral function oi x; hi, ^...A„; 
Ci, C2 ... Cn- The allotrious factor itself is made up of the product of squares 
of quantities all of the same form as the leading coefficient in DiX, which, 
from what has been shown above, is seen to be equal to 

{-■j^t{(CiC,...Ci)^(h„h...h)}. 

Hence each term in the continued fraction 

11 1 



(Aix + Bi)- (A,x + B,)-'"(AnX + B„)' 



57] of two Algebraical Functions. 579 

wliich is to be made equal to 



Co 

■+ — V + --- + 



x — hi X — h^ '" X — hn' 
is completely assigned in terms of x and the given quantities c and h. 

Art. (J3). The number of effective intercalations between the roots of 
^x, Fx is easily seen to be equal to the excess of the number of positive 
real numerators over the number of negative real numerators in the partial 
fractions of which -^ is the sum, and hence we see a priori, as an obvious 
consequence of a simple extension of the reasoning in Art. 47 [p. 515 above], 
that the inertia of the quadratic function 

2 j Ce (t(i + AflMs + hiUi + . . . + /ie"~Ht„)' ^_^ 

where Cs^-rnf-, will represent the value of the index in question. So too 

we may see that the formulae given for the residues to fx, f'x in Art. 46 
continue to apply to the residues Fx, (tx. That is to say, these residues 
when divided out by Fx will be respectively represented by the successive 
principal coaxal determinants to the matrix 

Co, Oj, O2 ■■•'^»«— 1) 

Si, S2, S3 ■■■S,n> 

O2 , O3 , O4 ... Sm+i J 



where in general 



Smi ^m+i •■■ ^2m— 2> 



X — hi X — hi X — tin 



and using the same matrix as above written with S' substituted for S, where 



8r = Ci (a; - hi) hi"" + Ca (« - A2) h„J+ ... +Cn{x- hn) hj, 

the successive principal coaxal determinants of the new matrix represent the 

successive denominators to the convergents of the continued fraction which 

<^x 

expre.sses „- . 

The expression for the numerators to the convergents may also, there is 
no doubt, be obtained by some simple modification (dependent on intro- 
ducing the quantities Cj, c^ ... c„) of the formula in Art. 41, p. [492]. 

I annex, more with the hope of suggesting than (in all instances) of 

conveying a full conception of the force of the definitions, a Glossary, or 

rather a Eepertory of the principal terms of art employed in the preceding 

pages, which might otherwise be apt to occasion some difficulty to persons 

unfamiliar with the subject. 

37—2 



580 On a Theory of the Syzygetic Relations [57 



Glossary of new or unusual Terms, or of Terms used in a new 
OR unusual sense, in the preceding Memoir. 

AUotrious. — The allotrious factor to a residue or quotient in the process of 
common measure applied to two algebraical functions is the constant factor of 
which such residue or quotient must be divested in order to become an integral 
and irreducible function. 

AjMcopated. — Applied to a type in the Theory of Cumulants, denotes a type the 
final or initial element of which has been taken away. If both are taken away, 
the type is said to be doubly apocopated. 

Bezoutic. — For definition of Primary and Secondary Bezoutics see first Section. 
Bezoutiant to two functions, each of degree n, is a homogeneous quadratic invarian- 
tive function of n variables, the form of which serves to assign the index of the 
scale of the effective intercalations of the real roots of the two given functions. 

Bezoutoid. — The Bezoutiant to two homogeneous functions obtained by dif- 
ferentiation from one homogeneous function of two variables. The Bezoutoid to a 
given function of m dimensions in the variables is accordingly a quadratic function 
of (m — \) variables, the form of which is sufficient for determining the number 
of real roots in the given function. 

Characteristic. — The employment of this word has been avoided in the pre- 
ceding memoir ; but as it contains an idea of capital importance in analysis, and 
especially in all inquiries of the kind here treated of, I subjoin the definition of 
its meaning. The characteristic of a simple condition of any kind is the rational 
integral function (in its lowest terms) whose evanescence necessarily and uni- 
versally implies and is implied by the satisfaction of such condition. A simple 
condition has always a single characteristic, abstraction being made of the alge- 
braical sign, which remains indeterminate. In like manner, a multiple condition, 
or a system of conditions, will have for its characteristic a plexus of rational 
integral functions, whose evanescence necessarily and universally implies and is 
implied by the satisfaction of such multiple condition or system of conditions. 
The number of functions in the characteristic plexus will however in general 
greatly exceed the index of the multiplicity of the conditions, and need not always 
be a unique system. There are however exceptions to this : thus the duplex 
condition, that a biquadratic function of x shall contain a cubic factor, or that a, 
curve of the third degree shall have a cusp, will each be definitely characterized 
by a plexus of two functions, and no more. 

The spirit of the higher analysis resides, and is to be sought for, in the logic 
of characteristics. 

Co-besoutiant. — Any homogeneous quadratic function similar in form and in 
its property of invariance to the Bezoutiant. 



57] of two Algebraical Functions. 581 

Cogredient and Gontragredient. — A system of variables is cogredient to another 
system when it is subject to undergo simultaneously therewith linear substitutions 
of a like kind, and contragredient when it is subject to undergo linear substitutions 
simultaneously therewith but of a contrary kind. 

Coiiibinmit. — A function of the quantities appearing in a given set of functions 
which remains unaltered as well for linear substitutions impressed upon the 
variables as for linear combinations of the functions themselves. 

Concomitmit. — Xomen generalissimum for a form invariantively connected with 
a given form or system of forms. 

Conjunctive. — A syzygetic function of a given set of functions. Any function 
which universally, and subject to no cases of excejytion, vanishes when a certain 
number of other functions all vanish together must be a conjunctive (that is 
a syzygetic function), or a root of a conjunctive of such functions. But if its 
vanishing is subject to cases of exception, then all that can be predicated of it 
is that it is syzyyetically related to such functions, but it may, and usually does 
happen, that it will be syzygetically related to them in more than one way. 

Coiitravariant. — A function which stands in the same relation to the primitive 
function from which it is derived as any of its linear transforms to an inversely 
derived transform of its primitive. 

Covariant. — A function which stands in the same relation to the primitive 
function from which it is derived as any of its linear transforms to a similarly 
derived transform of its primitive. 

Cumulant. — The denominator of the simple algebraical fraction which expresses 
the value of an improper continued fraction. See Type, infra. 

Determinant. — This word is used throughout in the single sense, after which 
it denotes the alternate or hemihedral function the vanishing of which is the 
condition of the possibility of the coexistence of a system of a certain number 
of homogeneous linear equations of as many variables. 

Dialytic. — If there be a system of functions containing in each term diflFerent 
combinations of the powers of the variables in number equal to the number of the 
functions, a resultant may be formed from these functions by, as it were, dissolving 
the relations which connect together the different combinations of the powers 
of the variables, and treating them as simple independent quantities linearly 
involved in the functions. The resultant so formed is called the Dialytic Resultant 
of the functions supposed ; and any method by which the elimination between two 
or more equations can be made to depend on the formation of such a resultant 
is called a dialytic method of elimination. In such method accordingly the process 
of elimination between equations of a higher degree than the first is always reduced 
to a question of elimination between equations which are of the first degree only. 

Discriminant. — -The resultant of the n differential coefficients of a homogeneous 
function of n variables. See Resultant, infra. 



582 On a Theory of the Syzygetic Relations [57 

Disjunctive. — A disjunctive equation is a relation between two sets of quantities 
such that each one of either set is equal according to some unspecified order of 
connexion with some one of the other set. 

Effective scale of intercalations is the series of the real roots of two functions 
of X written in order of magnitude after repeated processes of removing pairs of 
roots belonging to either the same function (when not separated by roots of the 
other function) : the roots of the two functions follow each other alternately. 

Effluent. — From every homogeneous function of any number i of variables of 
the degree mm', where m, m are any two integers, may be formed (as shown in the 
Calculus of Forms, Section II.) a covariantive function of the degree to and of yu 
variables, where /x is the number of permutations that can be obtained by dividing 
m' into i parts (zeros admissible), in which all the coefl&cients are numerical 
multiples of the given coefficients ; covariants so formed may be termed effluents 
of their primitive. An example of this occurs in the footnote to Section V., 
[p. 557], where the quantity there called § is a quadratic effluent of the Jacobian. 

Element. — A simple component of the type to a cumulant. See Cumulant, 
supra. 

Emanant. — The result of operating any number of times (suppose i times) upon 
a given homogeneous function of any number of variables x, y, z ... t with the 
operative symbol 

/ , d , d , d ^, d\ 

\ dx ^ dy dz dtj' 

is called the ith emanant of the function operated upon. Every emanant is a 
covariant to its primitive, the new variables x', y', z' ...t' being cogredient with 
the variables x, y, z ... t with which they are respectively associated. E^^+if, 
Eii+i4>, page [561], are emanants of / and (^. The process of emanation is one of 
incessant occurrence in the theory of invariants. When the order of the emanant 
is the same as the degree of the function (supposed to be rational and integral) 
from which the emanant proceeds, the form of the original function is repro- 
duced in the final emanant, the names only of the variables being changed. 

Endoscopic, Exoscopic. — When the coefficients of the functions concerned in 
any investigation are regarded as integral indecomposable monads, the method 
is called exoscopic, and endoscopic when the coefficients are treated with reference 
to their internal constitution as composed of roots or other elements. 

In addition to the examples in the footnote to Section I.*, these words have a 
marked and most important application in the theory of Invariants, especially 
of two variables. 

Porm. — Any function may be regarded as an opus operatum; the matter 
operated upon being the variables, and the substance of the operations being the 
form, which resides in the function as the soul in the body. A form is always 
common to an infinity of functions, but for greater brevity may be and frequently 
is called by the name of some specified function in which it is contained. 
[* p. 431 above.] 



57] of two Algebraical Functions. 583 

Fundamental. — The fundamental scale of a system of Invariants or Concomitants 
is a set of the same, whereof every other is a Rational Integral Function. 

Hessian or Hessean, named after Dr Otto Hesse, of Konigsberg (the worthy 
pupil of his illustrious master, Jacobi, but who, to the scandal of the mathematical 
world, remains still without a Chair in the University which he adorns with his 
presence and his name), is the Jacobian to the differential coefficients of a homo- 
geneous function of any number of variables. It is to a Jacobian what a Bezoutoid 
is to a Bezoutiant, or a Discriminant to a Resultant. 

Ml/per determinants. — See Memoir of Mr Oayley, Cambridge and Dublin 
Mathematical Journal, May 1845, and Crelle's Journal of about the same date. 

Imfro-per continued fraction is a continued fraction differing only from an 
ordinary one in the circumstance of negative signs being substituted for positive 
signs to connect the terms. 

Inertia. — The unchangeable number of integers in the excess of positive over 
negative signs which adheres to a quadratic form expressed as the sum of positive 
and negative squares, notwithstanding any real linear transformations impressed 
upon such form. 

Intercalations. — The theory of intercalations is the theory of the relative 
distribution of the real roots, or point-roots, of two or more equations, but in this 
theory the number of roots mutually interposed is to be taken only with reference 
to the number 2 as a modulus. 

Invariance. — The property (under prescribed or implied conditions) of re- 
maining invariable. 

Invariant. — A function of the coefficients of one or more forms which remains 
unaltered when these undergo suitable linear transformations. 

Inverse. — The inverse to a given square matrix is formed by selecting in its 
turn each component of the given matrix, substituting unity in its place, making 
all the other components in the same line and column therewith zero, and finally 
writing the value of the determinant corresponding to the matrix thus modified 
in lieu of the selected component. If the determinant to the matrix be equal 
to unity, its second inverse, that is the inverse to its inverse, will be identical, term 
for term, with the original matrix. 

Jacobian. — The Jacobian to n homogeneous functions of n variables is the 
determinant represented by the symmetrical collocation in a square of the n 
differential coefficients of each of the n functions. 

Kenotheme. — A finite system of discrete points defined by one or more homo- 
geneous equations in number one less than the number of variables contained 
therein. 

Limiting Series. — One set of quantities whose extreme values are exterior to the 
extreme values of a second set is set to limit the latter. 

Matrix. — A square or rectangular arrangement of terms in lines and columns. 



684 On a Theory of the Syzygetic Relations [57 

Minor Determinant. — Any determinant retained represented by a square group 
of terms arbitrarily chosen out of a matrix is a minor determinant thereto. The 
simple terms of the matrix are the last minors, and of course if the matrix is 
a square, it will itself in its totality represent a single complete determinant. 

Monotheme. — A line, or finite system of lines, defined by one or more homo- 
geneous equations two less in number than the number of the variables contained 
therein. 

Order. — The orders of a homogeneous function are the linear functions of the 
variables the least in number by aid of which the function admits of being 
expressed. 

Perayvimetrical. — A symmetrical matrix, in which all the terms in the diagonal 
bands transverse to the axis of symmetry are identical, is said to be persymmetrical. 
Example. An addition table. 

Quadrinvariant. — An invariant of which the terms are quadratic functions 
of the coefficients of the primitive. 

Relation (simple and compound). Vide Substitution, infra. 

Resultant. — The resultant of n homogeneous general functions of n variables 
is that function of their coefiicients which, equated to zero, expresses in the 
simplest terms the condition of the possibility of their coexistence. 

Rhizoristic. — A rhizoristic series is a series of disconnected functions which 
serve to fix the number of real roots of a given function lying between any 
assigned limits. 

Signaletic. — A signaletic or Semaphoretic series is a sequence of disjunctive 
terms, considered solely with reference to the algebraical signs of plus and minus 
which they respectively carry. 

Singular. — A proper algebraical function of a given degree, n, in one variable 
in its most general form, will, in respect to that variable, be of the nth degree 
in the denominator and the (w— l)th degree in the numerator, and will admit 
of being represented by a continued algebraical fraction of n terms, all of them 
linear. 

But for particular values of, or relations among, the coefficients entering into 
the given fraction this mode of representation fails, and the continued fraction, 
instead of consisting of linear terms n in number, will consist of terms, some of 
them at least, non-linear, and fewer than n in number. These then are the 
singular cases (or cases of singularity) in the theory of the development of an 
algebraical fraction under the continued fraction form ; and it will be seen that 
according to this definition the case of the development of any proper algebraical 
fraction in which the degree of the numerator is more than one unit below that of 
the denominator, belongs (strictly speaking) to the class of singular cases ; and 
this view of tlie case supposed is perfectly correct and conformable to the analogies 
of the subject. 



57] of two Algebraical Functions. 585 

Substitution (linear, siinilar or contrary). — A linear substitution is said to be 
impressed upon a system of variables when each variable is replaced by a linear 
conjunctive of all the variables. The matrix formed by the coefficients of sub- 
stitution arranged in regular order is called the Matrix of Substitution, and is of 
course a square. When two substitutions (impressed on two systems of variables) 
have the same matrix, they are said to be similar, and contrary when their matrices 
are contrarj', that is mutually inverse to each other. When two systems of 
variables are supposed to be subject to the condition that their substitutions 
are always similar or always contrary, they are said to be related or in simple 
relation, the relation being of cogredience in the one case and of contragredience 
in the other. 

When a linear substitution is impressed upon a system of independent variables, 
a corresponding linear substitution is necessarily impressed at the same time upon 
every complete system of homogeneous combinations (that is, products and powers 
and products of powers) of these variables, the matrix to which latter substitution 
will consist of terms which will be functions (depending upon the degree of the 
homogeneous combinations) of the terms of the matrix to the primitive substitution. 
This matrix may be termed a compound matrix, having the primitive matrix 
for its base. 

If, now, two systems of independent variables are subject to be synchronously 
impressed with substitutions, the matrices to which (not being both of them simple 
matrices) have for their bases matrices which are either similar or contrary, these 
two systems will be said to be in compound relation of cogredience in the one case, 
and of contragredience in the other. 

Syrrhizoristic. — A syrrhizoristic series is a series of disconnected functions 
which serve to determine the effective intercalations of the real roots of two 
functions lying between any assigned limits. 

Syzygetic. — A syzygetic function or conjunctive of a number of given rational 
integral functions is the sum of these affected respectively with arbitrary functional 
multipliers, which are termed the syzygetic multipliers. When a syzygetic function 
of a given set of functions can be made to vanish, they are said to be syzygetically 
related. 

Transform. — Equivalent to the French noun substantive " trans/ormee." 

Type. — The type of a cumulant is the series of the simple elements (or quotients), 
arranged in a fixed order, of which the cumulant is composed. 

Umbral. — The umbral notation is a notation according to which simple 
quantities are denoted by syllables, instead of by single letters (the composition 
of these syllables being governed by the mode in which the quantities which they 
express are obtained) ; and the single letters of such syllables are termed umbral 
quantities or wmhrce. 

Weight. — In this memoir (throughout the earlier sections) the weight of any 
quantity composed of the product of the coefficients of any given function or 



586 Syzygetic Relations of two Algebraical Functions. [57 

functions of x is used to denote the number of roots of x appertaining to the given 
function or functions which must be employed to express such quantity. More 
generally, when dealing with a system of homogeneous functions, the weight 
of a quantity may be defined with respect to any selected variable therein as the 
sum of the weights in respect to such variable of the several coefficients of which 
the quantity is composed (the weight of each several coefficient meaning the index 
of the power of the selected variable in that term of the given function or functions 
which is affected with such coefficient). These two definitions of weight may be 
perfectly well reconciled with each other by understanding the weight of a quantity 
formed from the coefficients of a function or system of functions of x to mean the 
weight, in respect to unity, of such quantity when the given functions are treated 
as homogeneous functions of x and 1. 

Zeta. — The symbol C (preceding a row of bracketed terms) is used to denote 
the product of the squared differences of the terms which it affects. 

[ ]. A bracket of this form, when enclosing a superior and an inferior row 
of terms m and n in number respectively, indicates the mn products of the 
differences obtained by subtracting each term in the second row from each term 
in the first row ; when enclosing an arrangement of terms in a single line, it is 
used to denote the cumulant of which such an arrangement is the type. 



58. 



ON THE CONDITIONS NECESSARY AND SUFFICIENT TO BE 
SATISFIED IN ORDER THAT A FUNCTION OF ANY NUM- 
BER OF VARIABLES MAY BE LINEARLY EQUIVALENT 
TO A FUNCTION OF ANY LESS NUMBER OF VARIABLES. 

[Philosophical Magazine, V. (1853), pp. 119 — 126.] 

In the Cambridge and Dublin MathematicalJournal for November 1850*, 
I defined an order as signifying anj' linear function of a given set of variables, 
and spoke of a general function of n variables as losing r orders when the 
relation between its coefficients is such that it is capable of being expressed 
as a function of (?) — r) orders only. It will be highly convenient to preserve 
the same nomenclature for the purposes of the present investigation. 

Dr Otto Hesse, in a long memoir in Crelle's Journal, the contents of 
which have been described to mef, but which I have not yet been able to 
procure, has given a rule for determining the analytical conditions for the 
loss of one order. I propose to give a more simple and comprehensive scheme 
of conditions than Professor Hesse appears to have discovered, applicable not 
to this case only, but to that of the loss of any number whatever of orders, 
and shall moreover show in what relation the substituted orders stand to the 
given variables. 

Dr Hesse's rule had been previously stated by me in the 4th section 
of my Calculus of Forms {Cambridge and Dublin Mathematical Journal, 
May 1852 1) as applicable to the case of a general function of the 3rd degree 

[* p. 171 above. ] ■ 

t A distinguished mathematical friend in Paris communicated to me with great admiration 
Professor Hesse's result overnight. I ventured to affirm that, 'to one conversant with the 
calculus of forms, the problem could offer no manner of difficulty. An hour's quiet reflection in 
bed the following morning, or morning after, sufficed to disclose to me the true principle of the 
solution. [Cf. Noether, Math. Annal. l. (1898) p. 138. En.] 

J Vide Vol. VII. p. 187 [p. 3.S5 above]. "When U represents a pencil of three rays meeting in 

a point, — =0, jt = 0, &a., and also therefore r = 0" (S and T being the two Aronholdian in- 
variants of U, and a, b, c, &c. the coefficients of U); "also in place of this system may he 
substituted the system obtained by taking all the coefficients of the Hessian zero." 



588 On Polynomial Functions which [58 

of three variables becoming the representative of three right lines diverging 
from the same point, which is the case of a cubic function of three variables 
becoming a function of two linear functions of these variables, that is to say, 
losing one order: this, perhaps, might have been noticed in the Professor's 
memoir. I gave also another rule for the same case; but the true fundamental 
scheme of conditions about to be set forth will be seen to embrace as mere 
corollaries all such and such-like rules, which in fact supply more or less 
arbitrary combinations of the conditions, rather than the naked conditions 
themselves in their simple form and absolute totality. 

I shall call the function to be dealt with JJ, and shall consider C/" to be a 
homogeneous* rational function of m dimensions in respect of a.'i, X2 ... x^, and 
shall inquire what are the conditions which must obtain when U is capable 
of being expressed as a function of only (?i — r) orders, say l^, l, ... In-r, each 
of which is of course a homogeneous linear function of the given n variables. 

Let the term derivative of U be understood to mean any result obtained 
by differentiating U any number of times with respect to one or more of the 
variables x^, x^... x^. The first derivatives will be of (m— 1) dimensions, 
the second derivatives of (m — 2) dimensions, and so on ; and finally, the 
(m— l)th derivatives will be homogeneous linear functions of «!, Xi...Xn. 
Suppose JJ to be expressible as a function of l^, l^ ... In-r- It is immediately 
obvious that the derivatives from the 1st to the (m — l)th inclusive will be 
all expressible as homogeneous functions of l^, l2...ln~r, and vanish when 
these vanish. But this statement is in substance pleonastic ; for by means 
of Euler's well-known law, any derivative of U, say K, may be expressed 
(to a numerical factor pr^s) under the form of 

dK dK dK 

X^ -J -f- X^ -J "r , . . -p Xji -~z , 

dx^ ax« axn 

and consequently, whenever the linear derivatives of U vanish, all the upper 
derivatives of U, including U itself, must vanish at the same time. The 
number of these linear derivatives, say v, will be the number of terms in 
a homogeneous function of n variables of (m — ,1) dimensions, that is 
to say, 

n(n — l) ... {n — m + 2) 
1.2... {m-l) ■ 

Again, if all the v linear derivatives vanish when the (n — r) equations 
li = 0, ^2 = . . . In-r = are satisfied, r being greater than zero, this can only 
happen by virtue of these v derivatives being linear functions of (n — r) 

* It is a common error to regard homogeneity of expression as merely a means for satisfying 
the desire for symmetry ; the ground of its application and utility in analysis lies, in fact, much 
deeper ; it is essentially a method and a power. 



58] admit of reduction in the number of Variables. 589 

of them. Now, conversely, I shall pi'ove, that if it be true that all the linear 
derivatives of U are linear functions (w — r) of them, then f/may be expressed 
as a function of these (n — r) ; and this rule, as will be immediately made 
apparent, will give the necessary and sufficient conditions for the loss of 
r orders in the most simple and complete form by which they admit of being 
expressed. For the proof of the rule, only one additional remark has to be 
made in addition to that already made, of the vanishing of the linear deriva- 
tives necessarily implying the simultaneous evanescence of all the other 
derivatives ; this additional remark being, that if the derivatives of any class, 
linear or otherwise, qud one set of variables, become all zero, the derivatives 
of the same class, qud any other set of variables linear functions of the first 
set and the same in number, will also become zero, for they are evidently 
expressible as linear functions of the first set. 

Now let d^, dz...dn-r he any {n — r) linear derivatives of V, of which 
all the other of the v derivatives of this class are linear functions, so that 
they vanish when these {n — r) vanish, and let U be expressed as a function 
of {d-i, d^... dn-r ; «i, ^2 . . . x^y Then we may write 

U = <^m,o + <f>m^i,i + ^7/t-2,2 + ••• +</>!, m-i + 4>o,m} 

where in general cf>m-e,e denotes a function homogeneous and of m — e dimen- 
sions in respect to rfj, c?2 •■• <^n-r, and homogeneous and of e dimensions in 
respect to «!, x^... os^. Now the linear derivatives of U all vanish when d^ = 0, 
(^2= ... d„_r= for all values of x-^, x^-.-oCr. Hence U=0 on the same 
supposition, and hence (^o,m is similarly zero. Also the first derivatives of U, 
qud dj, rfa ■•• dn-r, must vanish on the same supposition. Hence <f)i,m-i 
is identically zero; and so by taking the 2nd, 3rd... up to the (7?i — l)th 
or linear derivatives of U in respect to di, d^... dn-r, we find successively 
<p2,m-2t </>3,m-3 ••• <^m-i,i each identically zero, and consequently 

U = <j)m,o= (j)(di, d^... dn-r), 

as was to be proved. To express the fact of the v derivatives being linear 
functions of (n — r) of them, form a rectangular matrix with the coefficients 
of the V linear derivatives. This matrix will be n terms in breadth and v 
terms in depth. Let r= 1 : it is a direct consequence of the rule which has 
been established, that every full determinant consisting of a square n terms 
by n terms that can be formed out of this rectangular matrix must be zero : 
again, let r = 2 ; all the first minors, that is to say, all the determinants 
composed of squares (n— 1) terms by (n— 1) terms, must be zero, and so in 
general a loss of r orders will require that the (r — l)th minors shall all 
vanish; if r = w, the in - l)th minors, that is the simple terms of the matrix 
which are all coefficients of TJ, must vanish, or in other words, when the 
function is of zero order all the coefficients vanish (an obvious truism). 



590 



On Polynomial Functions which 



[58 



Thus, then, we see that the true rule for the loss of one order in a polynomial 
of any degree is precisely the same as the well-known rule for the loss of one 
order in a quadratic function ; the speciality in the latter case consisting 
merely in the fact that v being equal to n, the rectangular matrix becomes 
a square, and there is only one full determinant. Moreover, for any other 
value of r the above rule coincides with that given by me some time back 
in the Philosophical Magazine for the case of quadratic functions. 

Professor Hesse's rule for finding conditions applicable to the loss of one 
order is, as I have already stated, a consequence of the more simple scheme 
of conditions above given. It consists in forming the determinant 



d?U 



(PU 



d?U 



dxidx^ '" dxidxn 
d'U d'U 



d^U 



d'U 



d-'U 



dXndXi' dXndx^" dXn dx-n 



and equating the coefficients of this determinant fully developed separately 
to zero*. The attachment of the Professor to this particular form of covariant 
(I use the language of the calculus of forms) is readily intelligible, seeing 
the admirable application which he has made of it to the canonization of the 
cubic function of three variables, but it is really foreign to the nature of the 
present question ; the coefficients of this covariant may easily be shown to be 
merel}' the full determinants of the nx v rectangular matrix above described, 
or linear functions of these said determinants with numerical coefficients. 
Hence the ground of its applicability. 

Returning to the rule of the matrix, if we suppose the number of variables 
to be two, and call the coefficients of U 



ao, nai, ^w(?i— l)a2 ••• o^m. 



our rectangle becomes 



etc, 


«! 


Oi, 


02 


^2, 


as 


aj!_i 


an 



* A form capable of being so derived I have elsewhere termed (in compliment to M. Hesse) 
the Hessian of the function to which it appertains. This is the trivial name which is much 
needed on account of the frequent occurrence of the form, and has been adopted by Mr Salmon 



58] admit of reduction in the number of Variables. 591 

and the conditions become 

Cfotta — a^ = 0, tto ftj — a, fta = 0, 
Oia^ — a.2 =0 



a„_2a„ — a\_i = 0, a„_3a„ — «„_2a„_i = 0, &e., 

all of which equations are obviously true (when the function loses an order, 
that is to say, becomes a perfect power) and are satisfied (special cases 
excepted) when any (w — 1) independent equations out of the entire number 
obtain ; so that the number of conditions implied in the property to be 
represented is in exact conformity with the number of independent equations 
derived from the matrix, that is equations which, when satisfied, will in 
general cause all the rest to be satisfied. This conformity manifests itself 
also in the case of a quadratic function of n variables. But except in these 
two limitiog (and, in an occult sense, reciprocal*) cases of a function of 
two variables of the »th degree, or of the degree 2 and n variables, this con- 
formity in measure as the degree or number of variables rises, although it 
must substantially continue to exist, becomes, and in an accelerated degree, 
less and less apparent. 

Thus, take the simple case of a cubic function of three variables, and let 
us confine ourselves to the consideration of the conditions which must be 
satisfied when it loses a single order. Let TJ be written out at length, 

aa? + hy^ + c^ + 'ihyz^ + ^izx^ + ^jxy'^ + Sh'y'z + Si'z^o) + Sj'x^y + Qmxyz. 

in his admirable treatise on the higher plane curves. In systematic nomenclature it would be 
termed the discriminant of the quadratic emanant, or more briefly, the quadremanative dis- 
criminant. I have discovered quite recently that the long sought for symmetrical, and by far the 
most easy practical process for discovering the number of the real roots of an equation, is 
contained in, and may be deduced immediately from, a certain transformation of its Hessian I 

* There are frequent cases occurring in the calculus of forms of interchange between the 
•degree of a function and the number of variables which it contains. Thus, to select a striking 
example (although one where the interchange is not exact), the theory of the real and imagiuary 
roots or factors of a homogeneous function of two variables and of the mth degree may be shown to 
be immediately dependent upon the determination of the specific nature of a concomitant 
homogeneous function of the 2nd degree and of (n - 1) variables. For instance, if any ordinary 
algebraical equation of the 5th degree be given, a homogeneous quadratic function of four variables 
may be constructed, representing, consequently, a surface of the 2nd degree [the coefficients 
of which (as indeed is true whatever be the degree of the equation) will be quadratic functions 
of the coefficients of the given equation] ; and such that, according as the surface so represented 
belongs to the class of (1), impossible surfaces; (2), the ellipsoid or hyperboloid of two sheets; 
(3), the hyperboloid of one sheet ; the given equation will have 5, 3, or only 1 real root ! Moreover, 
an equality between two of the roots of the equation will be denoted by the loss of one order 
in the associated quadratic function ; and so many orders altogether will be lost as there are 
independent equalities existing between the roots. An entirely new light is thus thrown on 
M. Sturm's theorem ; and the number of real and imaginary roots in an equation is for the first 
time made to depend upon the signs of functions symmetrically constructed in respect to the two 
ends of the equation, which has long been felt as a desideratum. 



592 On Polynomial Functions which [58 

The matrix formed out of the coeSicients of the linear derivatives becomes 



a, j', 


i 


j, b, 


h' 


i', h, 


c 


m, h', 


h 


i, m, 


i' 


)'< j' 


m 



Now by the homaloidal law, if the terms in this rectangle were all unlike, 
the number of full determinants (3 terms by 3 terms) whose evanescence 
(except for special values) determines the evanescence of all the rest, should 
be (6 - 3 + 1)(3 — 3 4- 1), that is 4; but in the actual case, since the 
evanescence of all the full determinants is a necessary consequence of the 
function becoming a cubic function of two orders (that is, breaking up into the 
product of three linear functions of x, y, z), and as this decomposability, as is 
well known, implies only the existence of three affirmative conditions, the four 
full determinants 



a, j', i 




j, h, h' 




i', h, c 





a, j', i 




j, b, h' 




m, h', h 





a, J, * 


a, J , ^ 


j, b, h' 


j, b, h' 


i, m, i' 


j'> j, ™ 



* That is to say, a syzygetio relation must connect these four determinants. I may as well 
here repeat, that when the vanishing of a set of i rational integral functions necessarily, 
and without cases of exception, implies the vanishing of another rational integral function, 
then this function is termed a syzygetic function of the others ; and some power of it must be 
expressible under the form of a sum of i binary products of rational integral functions, one 
factor of each of which products must be one of the i given functions. When the vanishing of 
all but one of a set of functions in general necessarily implies the vanishing of that one, but 
subject to cases of exception for specific values of the variables, then it can only be affirmed 
that the functions of the set are in syzygy ; that is to say, that the sum of the products of each 
of them respectively by some rational integral function will be zero : the equation expressing 
this relation is termed a syzygetic equation. 

Thus, if we take the three full determinants that can be formed out of the matrix 

a, a, 

b, ft 



that is a;8 - 6a, by - c^, ea - ay, 

these are in syzygy, for we can form the equation 

c (a(3 -ba) + a{by-c^) + b {ca - ay) = 0. 
This, however, is not the only equation of the kind that can be formed, for 

y {a^ -ba) + a {by - C;8) + /3 {ca -ay) = 
is a,lso identically true. We see in this case that the evanescence of any two of the three functions 



58] admit of reduction in the number of Variables. 593 

which in the general case would be entirely independent, in this case cease 
to be so ; and the vanishing of three of them must draw along with it by 
necessary implication (except for special values) the evanescence of the 4th, 
for thus only can the necessary conformity between the number of affirmative 
conditions and the numiber of unimplicated equations come to take effect. 
The clear and direct putting in evidence of this peculiar species of implication 
demands and deserves to be minutely considered ; and as it must in part 
borrow its explanation from the very little yet known of syzygetic relations, 
so it must also throw new light on that great and important, but as yet 
unformed and scarcely more than nascent theory. 

In conclusion, it is apparent from the demonstration above given, that 
when U, a function of n variables, becomes expressible as a function of 
{n — r) orders, these orders may be taken respectively any independent linear 
functions of the linear derivatives of U, which remark completes the theory 
of functions subject to the loss of one or more orders. It is obvious (and I 
am indebted to my esteemed friend Mr Cayley for the remark), that the 
conditions furnished as above by the (m — l)th, that is linear derivatives, 
are identical with and may be more elegantly replaced by those involved 
in the assertion of the existence of linear relations between the 1st or 
(m — l)th degreed derivatives, and we have then this very simple rule ; 
if (f), a function of x-^, w„ ... Xn, is expressible as a function of n — r linear 
functions of x^, x„ ... Xn, it is necessary and sufficient that r independent linear 
relations shall exist between 

d(j) d<j) d4> 

dx-i ' dx^ ' ' ' dxn ' 

a^-ba; 67 - c/3 ; ca- ay will in general imply the third, subject, however, to special cases of 
exception. Thus, if the 1st and 2nd vanish, the 3rd must vanish unless b and /3 both vanish ; 
if the 2nd and 3rd vanish, the 1st must vanish unless c and 7 both vanish ; if the 3rd and 
1st vanish, the second will vanish unless a and a both vanish. It will thus be seen that a 
peculiar species of astricted syzygy obtains between the three proposed functions, which enables us 
to affirm that in general, and except under extra special conditions, all three must vanish simul- 
taneously. If two out of the three vanish, and the 3rd does not vanish, it is not merely (as might 
at the first blush of the theory of syzygy be conjectured) because some one other function vanishes 
in its place, but necessarily because a plurality of entirely independent functions (two simple letters 
as it happens here) each separately vanish. Thus we see how all but one of a set of functions 
Xii X2 ■•• Xn ™^y "' general, and yet not universally, necessarily vanish when all the rest vanish : 
to say that one syzygetic equation such as 

XiXi' + X2X2'+--+X„X„' = 
obtains, is not enough to explain the circumstances of the case ; the fact is, that several distinct 
systems of values of Xi'i X2' ■•• Xn ^^^^ ^^ found capable of satisfying the equation, so that each 
of the functions xi' X«--Xn '^'^1 have a system of syzygetic factors attached to it, and these 
unrelated, in the wide sense that, if we take Xn'i Xn "i ^''y '^^ °f '^^^ syzygetic factors attached 
to Xn > tlifiy ''^ill ""* ^^ *™ ^y^ysy with xi i X2 ■ • • Xn-i ; ^° t^aX when these [n-1) functions vanish, 
the vanishing of Xn ^"^ Xn" represents two distinct and completely independent conditions. 
Thus, in fine, the mutual implication of functions will in general denote the possibility of forming 
a series of syzygetic equations between them, — a remark, this, of no minor importance. 

s. 38 



694 



On Polynomial Functions. 



[58 



This rule itself also, it is evident, is capable of an independent and 
immediate demonstration by means of integrating the partial differential 
equation or equations by which it admits of being expressed. The above 
theory may readily be extended to functions of several systems of variables. 
Thus, for instance, the determinant 



h, 



a , 



vanishing will be indicative of the function 

{a xu + b wv + c ocw^ 
+ a'i/u+ b' yv + c' yw\' 
+ a"zu + h"zv + c"sw) 
being linearly equivalent to a function of the form 
Ax'u' + Bx'v') 
. + Cy'u' + Dy'v) 

that is losing an order in respect of each of the two systems x,y,z; u, v, w; 
and so in general. 



59. 



ON Mr CAYLEY'S IMPROMPTU DEMONSTRATION OF THE 
RULE FOR DETERMINING AT SIGHT THE DEGREE OF 
ANY SYMMETRICAL FUNCTION OF THE ROOTS OF AN 
EQUATION EXPRESSED IN TERMS OF THE COEFFICIENTS. 

[Philosophical Magazine, v. (1853), pp. 199—202.] 

For a considerable time past, among the few cultivators of the higher 
algebra, a proposition relative to the theory of the symmetrical functions 
of the roots of an equation has been in private circulation, which, to say 
nothing of the important applications of which it has been found susceptible 
to the calculus of forms, merits (by reason of its extreme simplicity), although, 
strange to say, it has, I believe, not yet obtained, a place in elementary 
treatises on algebra. The proposition alluded to I have reason to think 
first came to be observed in connexion with my well-known formulie for 
Sturm's auxiliary functions in terms of the roots given in this Magazine. 
The theorem is briefly as follows. If a, b, c, &c. be the roots of an equation 

a:" + J*!*"-' + p^x^-^ + &c. = 0, 

any symmetric function such as 2a°6^c>..., where a, /3, y ... are positive 
integers arranged according to the order of their magnitudes in a descending 
(or, to speak more strictly, non-ascending) order, when expressed as a function 
of the coefficients, will be made up of terms of the form p^^^ f^-pi'' ■ . . Pk^i', such 
that ^1-1-^2-1-^3-1- ... 4- ^fc will be equal to a for some terms, but will for no 
term exceed a ; a being, as above described, that one of the indices a, /3, 7 . . . 
which is not less than any of the others. 

I had prepared, and indeed despatched, a somewhat elaborate proof 
of this theorem for the Cambridge and Dublin Mathematical Journal ; but 
on proceeding to explain my method to Mr Cayley, elicited from that sagacious 
analyst the following excellent impromptu, which I think too valuable to be 
lost ; and as it is now a twelvemonth or two since our conversation on the 
subject took place, and the author has not cared to put it on record, I feel 

38—2 



596 On cm Impromptu Demonstration of Mr Cayley. [59 

myself under an obligation so to do, the more so as it entirely supersedes the 
comparatively inelegant demonstration of my own which I had previously 
intended to publish. 

The method rests essentially on the following well-known theorem given 
by Euler relative to the partition of numbers ; to wit, that the number of 
ways of breaking up a number n into parts is the same, whether we impose 
the condition that the number of parts in any partitionment shall not exceed 
m, or that the magnitude of any one of the parts shall not exceed m. 
Of this rule more hereafter — for the present to its application to the matter 
in hand. 

Since a,h, c ... are the roots of «" + 'piX^~'^ + ... , we have 

Pi=a+h + c + ... 

P2 = a6 + ac + 6c + . . . 

Ps = abc + abd + acd + ... 



Let a-|-/3+7+ ...=n, none of the quantities a, /8, 7... being greater 
than m, but a, /3, 7... being otherwise arbitrary and capable of becoming 
equal to any extent inter se. Also let X + fi + v + ... = n, the number of 
quantities X, fi, v, Sic. being never greater than m, but the quantities 
themselves being otherwise arbitrary, and being capable of becoming equal 
to any extent inter se. By Euler's rule the number of systems a, /3, 7 . . . is 
the same as of the systems X, ft, v..., say P for each. For any system 
X, /t, 1/ ... , we shall h.a,ve pi^p^p^ ... , by virtue of the equations above written, 
expressible as the sum of terms of the form %a'^W c* ... ; it may easily be 
made ostensible, that all the combinations of a, /3, 7 ... subject to the above 
prescribed conditions must come into evidence by giving X, fi, v ... all the 
variations of which they admit ; but this is also immediately obvious indirectly 
from the consideration, that were it otherwise, linear relations would subsist 
between the different values of p^p^p^..- , which is obviously absurd. Hence, 
then, we shall be able to express the P quantities of the form p^p^... by 
means of linear functions of the P quantities Sa" 6^ c^ . . . ; and conversely, 
by solving the linear equations thus arising, the P quantities Sa° 6^ c^ ... 
may be expressed in terms of the quantities p^p,,...; consequently Sa^t^c^ ..., 
where m is greater or not less than any of the quantities /3, 7 . . . , will be 
expressible by means of combinations p^Pij...., where the number of co- 
efficients PkP^... (any number of which may become identical) is for some 
of the combinations as great as, but for none of the combinations greater 
than m, as was to be proved. It will of course be seen that, for the 
purposes of the demonstration above given, it would have been sufficient 



59] On an Impromptu Demonstration of Mr Cayley. 597 

to have been able to assume that the number of partitions, when the greatest 
part is not allowed to exceed m, is not greater than the number of partitions 
when the number of parts in any one partitionment does not exceed m. 
The equality of these two numbers would then evince itself in the course of 
the demonstration as a consequence of this assumption. 

A word now as to Euler's beautiful law upon which the above demon- 
stration is based. 

A corollary from it, obtained by subtracting the equation which it gives 
when the limiting number is taken (m — 1) from the equation which it gives 
when the limiting number is m, will be the following proposition. The 
number of modes of partitioning n into m parts is equal to the number 
of modes of partitioning n into parts, one of which is always m, and the 
others m or less than m. This proposition was mentioned to me by 
Mr N. M. Ferrers*, whose demonstration of it (probably not different from 
that of Euler's for the other proposition, of which it may be viewed as a 
corollary) is so simple and instructive, that I am sure every logician will be 
delighted to meet with it here or elsewhere. It affords a most admirable 
example of that rather uncommon kind of reasoning whereby two abstract 
integers are proved to be equal indirectly, by showing that neither can be 
greater than the other. 

If there be a group of A's and a group of B's, and every A can be shown 
to produce a B, and every B can be shown to produce an A, no matter 
whether the A producing a S is the same as, or different from, the A 
produced by that B, it is obvious tha,t the number of A's cannot exceed that 
of the jB's, nor of the jB's that of the A's, and the two numbers will therefore 
be equal. 

Take any such grouping as 3, 3, 2, 1, say A. This may be written as 

1, 1, 1 

1, 1, 1 

1, 

and by reading off the columns as lines, may be transformed into the group 
1, 1, 1, 1 
1, 1, 1 

1, 1 

that is 4, 3, 2, say B. 

* I learn from Mr Ferrers that this theorem was brought under his cognizance through a 
Cambridge examination paper set by Mr Adams of Neptune notability. 



598 On an Impromptu Demonstration of Mr Cayley. [59 

In A the number of parts is 4. In B the greatest part is 4; the others 
might be (although they happen not in this particular instance to be) 4, but 
cannot be greater than 4. And so every A in which the number of parts 
is 4 will give rise to a £ in which 4 is one of the parts, and every other part 
is 4 or less, and evidently (although, as above remarked, this is immaterial 
to the demonstration) every such B gives reciprocally the same A from which 
it is itself derived ; hence the number of A'a and Bs is equal. This is the 
theorem which, for the sake of distinction, I have called the Corollary to 
Euler's. Euler's own is proved by the same diagram ; for if we define A 
as a grouping where the number of parts does not exceed 4, we get a definition 
of jB as a grouping where the greatest part does not exceed 4, and so in 
general. We see that this theorem may be varied also by affirming that the 
number of ways in which n may be broken up, so that there shall never 
be less than m parts, is the same as the number of ways in which it may 
be broken up into parts, the greatest of which in any one way is not less 
than m. So, again, a similar diagram makes it apparent, that if we break 
up each of i numbers into parts so that the sum of the greatest parts shall 
not exceed (or be less than) m, the number of ways in which this can be done 
will be the same as the number of ways in which these i numbers can be 
simultaneously partitioned so that the total number of parts in any simul- 
taneous partitionment shall never exceed (or never be less than) m ; and 
doubtless an extensive range of analogous general theorems relative to the 
partitioning of numbers may be struck out by aid of the same diagram, 
by no meaas easily demonstrable unless this simple mode of conversion happen 
to be thought of, but in that event becoming intuitively apparent. This 
mode of conversion is precisely that (only applied to a more general state 
of things) whereby, in elementary arithmetic, it is established that m times 
n is the same as n times m. A consideration of the process by which 
the mind satisfies itself of the universality of this law, has been always 
sufiicient to convince me of the absurdity of ascribing to an inductive process 
the capacity of the human mind for forming general ideas concerning 
necessary relations. 



60. 



A PROOF THAT ALL THE INVARIANTS* TO A CUBIC 
TERNARY FORM ARE RATIONAL FUNCTIONS OF ARON- 
HOLD'S INVARIANTS AND OF A COGNATE THEOREM 
FOR BIQUADRATIC BINARY FORMS. 

[Philosophical Magazine, V. (1853), pp. 299—303, 367—372.] 

Although contrary to the order of exposition indicated in the title to 
this paper, I shall, as the simpler case, begin with establishing the theorem 
for a biquadratic form, say F in cc, y. Let 

F = ax^^- %!x?y + locx'^y"- + Mxy^ + ey\ 

s = ae — ibd + 3c', 

t = ace — ad" — c^ — Jfe + 2bcd, 

s and t are the two well-known invariants of F. I propose to prove that 
there can exist no other invariants to F except such as are explicit rational 
functions of s and t. 

Let F, by means of the substitution of /« + gy for x, and f'x + g'y for y, 
be made to take the form f^ = x'^ + y* + %mx-y"'. Then by the characteristic 
property of invariants, if / (a, h, c, d, e) be any invariant to F of the degree 
q, we must have 

7 (1, 0, VI, 0, 1) = {fg' -fgy I {a, h, c, d, e) ; 

and it will be sufficient to prove that /(I, 0, m, 0, 1), or say more simply 
I (m), can only have the two radically distinct forms corresponding to 
s and t, that is 

(s) = 1 — Sm" and (t) = m — m^, 

any other admissible form of / being a rational explicit function of these two. 

* A Constant in analysis is any quantity which in its own nature, or by the explicit conditions 
to which it is subjected, is incapable of change. An Invariant is an expression apparently liable 
to change, but which, owing to certain compensations in the modifying tendencies impressed upon 
it, remains as a whole unaltered. The former may be compared to a fixed point or system in 
mechanics ; the latter to a point or system free' to move, but kept at rest under the combined 
operation of contending forces. 



600 On AronlioMs Invariants. [60 

It may be shown* that the parameter m in /i will have six different 
values and no more. In the first place, if we write ix for x in _/i (i meaning 
V— 1), it is obvious that m becomes — m. Again, let x + ty and x — ly be 
substituted in place of x and y respectively; then calling (/) the value 
assumed hyfi, when this substitution is made, 

(/) = (« + tyY + (« - lyY + 6m (x^ + y^y 



Hence if we write 
and 



= (2 + 6m)^xf^ + f + 6j^a^fY 



(2T6^i'"+(2+6m)i2/f°^^- 



and call what/i becomes after these substitutions /a, 

/a = a^ H- 2/^ + 67 (m) x^, 
-y(m) denoting :^i±^. 

In like manner, by writing in /a 

1 , t 

12+67 (m)}i "^^ "^ {2 + 67 (m)}i ^ "'' 

1 t 

{2 + 67(m)}i* {2 + 67(m)ji^*°''^' 

/s = «■* + 2/^ + 67^ (m) «'i/", 

-1+-^ + "^ 
„ , . 1 + 3m -2- 2m -1-m 

72 (ni) = 



and 

we obtain 
where 



_ — 1 + m — 2+ 6m — 1 + 3m ' 

^+^TT3^ 

7 (?n) is a periodic function of m of the third order, for we find 

,/ N of / M -(l + 3m)-(-l+m) 
rys (m) = 7= \y(m)\ = — f^ ^ — r — =r7 — ; t = m. 

It will of course be observed, also, that 

^2 ^jj^) = _ ry (— m) and 7 (m) = — 7^— m). 

* See Addendum [p. 607 below]. 



60] 



On AronhoMs Invariants. 



601 



Hence 

(_^)(_^)(m,) = -y(-m) = »tt, {-'^"■){-r^-){m) = -'f{-m) = m. 
So that, in fact, the six values of the parameter are 
m, 7 (m), 7^ (m), 
— m, — 7 {m), — 7^ (m), 

forming two cycles, having the remarkable property that the terms in the 
same cycle are periodic functions of the third order of one another, and each 
term in one cycle is a periodic function of the second order of every term 
in the other cycle. 

The modulus of substitution for passing from /i to f^, that is the square 
of the determinant 

1 I 



(2 + 6m)i' (2 + 6m)i 

1 -t 

(2+6m)i' (2 + 6?n,)i 

(- 2t)' - 2 



^'^ 2 + 6m ' 1 + 3m ' 

So that if / (m) be the value of any invariant of the degree q, corresponding 

to the form/i, and consequently / [ ^ ^ ] the same for/^, we must have 






)• 



In like manner, by means of /s it may be shown that we must have the 
further equation 



- 2 y VI - 3my 

These equations are easily verified for the values of (s) and («). 

Thus 

(l + 3m)°- L ./m-iy 



''^^M^)l 



(t) = m - 



(1 + 3m)^ f m - 1 
3m + 1 



f m—1 Y 
^3m + l 



(1 - 3m)' ( m + l f m + l \ 



■ omf III, -t- ± / ■//(, -r J- \ I . 
8 |l-3m ~ \\-Zm) \ ' 



602 On Aronhold's Invariants. [60 

and it is moreover obvious, that the values of (s) and (i) might have been 
found a priori by means of these functional equations. 

The essential point of inference for my present purpose from the equations 
above, v?hich are of the form 

./m — IN „ J-/771 + 1' 



I {m)= H X I (^^^^^ = K X I 

^ ' \3m+l/ 



l-3m 



is this, that if /(m) contain any power of m, say m', it must also contain 
(m — 1)' and (m + 1)' ; in a word, (m' — m)', which, by the way, it may be 
noticed, is (i)'. Now, if possible, let there be any invariant Iq {771) of the 5th 
degree in m which is not a rational function of (s) and (t). If we make 
ix + Sy = q, as many integer solutions as exist of this equation (in which zero 
values of x and y are admissible), so many functions of the form (s)^(<)^ may 
be formed of the degree q in m, and all of them of course invariantive 
functions. 

As regards the general nature of any invariantive function in m, since 
the change of x into —x in oc^ + y* + Gmx^y^ introduces no change into the 
invariant if q be even, but changes the sign if q be odd, it follows that Ig (ni) 
is of the form (p {w?) when q is even, and of the form m<p (vi^) when q is odd. 

Let fi be the number of solutions of the equation in integers above 
written. Then, by linearlj' combining all the different values of (s)" {t)v with 
Iq (m), it is obvious that we may form a new invariant, say I'g , in which the 
fj, first occurring powers of m will be wanting, that is in which the indices 
0, 2, 4 . . . (2fi — 2) will be wanting when q is even, and 1, 3, 5 . . . (2)u. — 1) when 
q is odd. Hence in the former case the new invariant will contain m-'', and 
in the latter case 7w^^+^ ; and therefore, by virtue of what has been shown 
already, I'g will contain (m^ — i7if>^ in the one case and (m^ — 7?i)'^"'"^ in the 
other. 

Firstly, let q = 6i, or Qi + 2, or 6i + 4 ; then /u. = i + 1 ; and therefore 
(m^ — mf^*", which is of the degree 6i + 6 in m, is contained as a factor 
in / which is of the degree q only, a quantity less than 6i+6, which 
is absurd. 

Again, secondly, let q=Qi+ 1, then iJi, = i; and (m' — TTifi^'^'^ is of the 
degree 6i + 3 in m, and is contained as a factor in /, which is of the degree 
Qi + 1, which is again absurd. 

Finally, if ^ = 61 + 1, or 6i + 3, /x = i+l; and the factor (m^ — m)*+^ is of 
the degree 6i + 9, that is, in each case, greater than q, which is absurd, 
and thus the theorem is completely demonstrated. 

It may for a moment be objected, that we have been dealing only with 
a particular form x^ + Gmafy^ + y', instead of the general form 
ax* + 4<bcc'y + 6cofy- + 4<dxy^ + ey* ; 



60] On Aronhold's Invariants. 



el? 



but the latter is always reducible to the former by means of a definite linear 
substitution ; and if we call the modulus of the substitution, that is the 
square of the determinant formed by the coefficients of substitution, M, 
to every general invariant /g of the jth degree, to the latter corresponds 
a partial form (/,) of invariant to the former, such that 

and consequently, since every (J) is a rational function of (s) and (t), so must 
every / be the same of s and t; unless, indeed, it were possible to have 

/,. = j^, (/,), q being different from and greater than q : but if this were the 

case, since 1^ = -^ (Z,), a power of M the modulus would necessarily be an 

invariant ; but in passing from oi* + y* + ^mx^y"- to sc^ A-y^ + 67 {m) x^y", 1 + 3m 
becomes the modulus, which we know is not an invariant. Hence the 
proposition is completely established for the case of the biquadratic function 
{x, yf*. 

Now let us proceed to Aronhold's famous S and T, the invariants to the 
general cubic function {x, y, zf, forms equally dear to the analyst and 
geometer. ( Vide Mr Salmon's Higher Plane Curves passim.) 

The method will be precisely the same as that applied to s and t ■\. 

We commence with the canonical form 

a? -\- y^ + z^ + &inxyz. 

On substituting xJry + z, x + py + p^z, x + p-y + pz for x, y, z, where p is the 
cube root of unity, the above quantity takes the form 

(3 + 6m) {«=> + 2/' + ^' + 6/3 (m) a;?/.^}, 
where 

18 -18m _ 1-m 
^^™'' "6(3 + 6m) ~l+2m' 

a periodic function in m of the second order only, for 

1 + 2m — 1 + m 



/3= (m) = 



1 + 2m + 2 - 2m 



* I have made a tacit assumption throughout the foregoing demonstration (which is, liowever, 
capable of an easy proof), namely that if any fractional function of the coefficients of any 
form be invariantive, the numerator and denominator must be separately invariants. 

+ The s is Mr Cayley's property, the t belongs to Professor Boole, having been by him imparted, 
in the infancy of the theory, to Mr Cayley, by whom it was first given to the world, at least in its 
character as an Invariant. 



-/04 On Aronhold's Invariants. [60 

But if we write for x in the original form px, it becomes 

a? + lf-\-z^ + Qpmxyz ; 
and if for x we write p'^x, it becomes 

a? + -tf + z^ + Qp-mxyz. 

Hence we can by linear substitutions obtain from x^ + y^ + z^ + Qmxyz the 
three additional forms 

ay' + y^ + z' + 6^ (m) xyz, 

w^ + y'^+ z^ + 67 (m) xyz, 

id' + y^ + z^ + QS (m) xyz, 
where 

„,, 1— ?re .. „1— pm p^ — m 

^^™> = rT^' -^^"'^^p-i^^hr^p^' 

^ ^ '^ 1 + 2|0=m 1 + 2p^m 

In all, there will be twelve values of m forming three remarkable compound 
cycles, 

m, /3 (?»), 7 (m), 8 (wi), 

pm, p^ (m), /37 (m), pS (m), 

jO-m, p-^{m), p-y{m}, p-S (m). 

It would be beside my present object to seek to develope fully the 
functional relations in which the several terms of these cycles stand to one 
another : the interesting relations 

j8^ (m) = 7^ (m) = 8'' (m) = m, 

^y (m) = 7^ (m) = 8 (m), 

ryB (m) = S7 (m) = /3 (m), 

S^(m) = /3S(w) = 7(m), 

have been already* stated by me in another place (Cambridge and Dublin 
Mathematical Journal, March 18.51f). 

The (*S) of the canonical form corresponding to the S of the general form 
is m — m*; and the (T) corresponding to the T of the general form is 
1 — 20m' — 8m''. (See my Calculus of Formsj, Cambridge and Dublin Mathe- 
matical Journal, February 1852.) It is my object to show that any other 
invariant (/) to the canonical form must be a rational function of S and T. 

In the first place, I observe that every invariant to any function of an 
odd degree i of any odd number p of variables must be of even dimensions ; 
for if the degree of the dimensions be q, and D the determinant of the 

[* p. 192 above.] f Vide Addendum [p. 607 below]. [J p. 311 above.] 



60] On Aro7ihold's Invariants. 605 

coefficients of substitution, the invariant to the transform becomes the original 

i« in 

invariant affected with a factor JD'' , where — must be an even integer, since 

otherwise the sign of this multiplier would be equivocal and indeterminable ; 
hence when i and p are both odd, q must be even. Thus, then, /(m) in the 
case before us must be an even -degreed function of m. Moreover, since the 
change of x into px converts in into pm, and Iq{in) into pilq{m), for D becomes 
p when X, y, z become px, y, z, Iq{m) must be of the form <^{ni'), m?<p(Tn^), 
7ii(f> (m^), according as the index q is of the form Qi, 6i + 2, 6i + 4. 

By precisely the same reasoning as was applied to the preceding case of 
(s) and (t), we see that any invariant of m which contains m' must also 
contain (1 — m)^ (1 — pmy, (1 — p-vif, that is must contain (m — m^y, which 
in fact is {Sf. If, now, we consider any invariant of the 5th degree in 
m, I {m\ and suppose it to be other than a rational function of (*8) and {T), 
and if we take /a to denote the number of the solutions of 4a; + 6y = g, 
it will follow that we may form an invariant I' (m), which, when q is of the 
form I2i or 12i + 6, will contain m, and consequently (m — m^)'''+^ as a factor ; 
and in like manner when q is of the form 12i + 2 or 12i + 8, will contain 
(m — m*y'^+^ as a factor ; and when q is of the form 12i + 4 or 12i + 10 will 
contain {m — m'')^''+^ as a factor. Now when 



+ 1. 

+ 1; 



+ 1: 



q = 12i, /j, = i 

q = 12i + G, p, = i 
when 

q=12i + 2, n = i: 

g = 1 2t + 8, /j, = i 
when 

g' = 12i + 10, /x = i + l, 

g = 12i + 4, /A = i + 1. 

Hence the factors dividing Iq in these several cases will be of the respective 
degrees 

12i+12, 12i+12; 12i + 8, 12x + 12; 12i + 16, 12i+16; 

corresponding to q, being of the several values 

12i, 12^+6; 12i-t-2, 12t + 8; 12i + 10, 12t + 4; 

which is clearly impossible. This proves the theorem in question (the 
passage being made from the canonical to the general form, as in the former 
part of this investigation), to wit, that S and T form what I have elsewhere 
termed a fundamental scale of invariants to the cubic ternary form, entering 
as the exclusive ingredients into every other invariant that can be derived 
from such form. 



606 On AronholcTs Invariants. [60 

A word of waruiiig is necessary before I lay down my pen : that there can 
be only two algebraically independent invariants to {x, y)* or {x, y, zf, is an 
immediate consequence of the canonical form of each having but one para- 
meter ; so in general there can be at most but (n — 2) absolutely independent 
invariants of {x, yY ', but the point established in the preceding investigation 
goes to show that there can exist no other invariants than such as are 
rational functions of s and t in the one case, and 8 and T in the other. I 
shall take some other occasion to establish a similar conclusion for the forms 
{x, yY and (x, yf. 

I have shown that there exist three invariants to the one of the degrees 
4, 8, 12, and four to the other of the degrees 2, 4, 6, 10; and I shall demon- 
strate that any other invariant to either form must be a rational function of 
those above stated. For the cubic form {x, yf we know that there is but one 
invariant, namely its discriminant. Thus, then, for n = S, n = 4i, n = 5, n = 6 
the number of absolutely independent invariants is n — 2, and the number 
of linearly independent invariants is no greater. But this result is by no 
means generally true. It may be proved by means of a great law of 
reciprocity* which I myself originated, but unfortunately threw aside, and 
which M. Hermite has since demonstrated, that there are more than five 
linearly independent invariants to (x, yj, and more than ten, in fact twelve 
at least, to {x, yY^ ; that is to say, it is impossible in the latter case to 
find ten of which all the rest shall be rational functions, although an 
algebraical equation connects any 11. So, again, if we take a system of two 
cubic equations, there are only five absolutely independent invariants ; but 
there are not less than seven linearly independent fundamental invariants, 

* The theorem of reciprocity alluded to in the text is the following : — If to any function 
(x, ?/)" there exists an invariant of the order m in the coefficients, then to (x, 2/)™ there exists 
an invariant of the order n in the coefficients ; or more generally, which is M. Hermite's 
addition, if to any system of functions (x, j/)",, (x, 2/)"2 ... [x, y)H there exists an invariant of the 
several dimensions m^, m, ... ?»i in the respective sets of coefficients, then conversely to a system 
(x, 2/)™i, (x, y)'"2 ... (x, !/)"'i there exists an invariant of the dimensions 71^, 71^.. .n^ in the respective 
sets of coefficients. 

I had previously shown in this Magazine [p. 279 above], that Mr Cayley's formulee for finding 
the number of biquadratic invariants to any function (x, j/)", given in that remarkable paper 
of his on linear transformations [Cayley's Collected Papers, Vol. i., p. 95], where first dawned 
upon the world the clear and full-formed idea of invariants (the most original and important in- 
fused into analysis since the discovery of fluxions), could be expressed by means of the number of 
solutions of the equation in integers 2x -)- 3j = n, the square of the quadratic invariant (which only 
exists for even values of n) counting for one in the fundamental biquadratic scale ; this is of course 
a direct consequence, through the law of reciprocity, of the fundamental scale to (x, yy consisting 
of a quadratic and a cubic invariant. My discovery of the fundamental scale of invariants to 
(x, yy and (x, y)^ now enables us, through the same law of reciprocity, to express the number of 
distinct Quintic and Sextic invariants to (x, ?/)", namely as being the number of integer solutions 

ot X + 2y + Zz=-7 in the one case, B-ni o{ x + 2y + Sz + 5t = - in the other. 



60] On Aronhold's Invariants. 607 

of which any other invariant must be a rational function. In fact, if we 
take for our two cubics 

U = aa? + Zhx^y + '?>cxy- + dy^, 

V=ixx^-\- 2/3x^y + Syxy' + Sy^, 

the five coefficients of the powers of X in the discriminant of U + \V, each of 
which is of four dimensions in the two sets of coefficients combined, are all 
invariants of the system ; but there will be besides two more, one of which 
is a Combinaut of six dimensions, being the resultant of U and V; the other 
is a Combinant of two dimensions only, namely aS — Sby + 3c/3 — da. These 
seven together form the fundamental constituent scale. 

The two last-mentioned may be expressed algebraically (by the introduction 
of square roots) as functions of the other five, but of course not as rational 
functions of the same. My attention was more particularly called to the 
search of a proof of the completeness of the Aronholdian system of invariants, 
by an inquiry as to the possibility of rigidly demonstrating that there could 
exist no others not made up of these, addressed to me in the spring of last 
year by one of the most gifted geometers of this or any other country. A 
morning or two after the inquiry reached me, in a walk before breakfast by 
the side of the ornamental water in St James's Park (a time and place by no 
means, according to my experience, unfavourable to the inspirations of the 
analytic muse), I had the satisfaction of falling upon the rather piquant 
demonstration above given, which essentially rests upon a principle, requiring 
no harder exercise of faith than the concession of the impossibility of a 
greater being contained in or proceeding out of a less. 



Addendum. 

On the nature of the three Cycles of four terms each which contain the twelve 
values of the parameter to the canonical form of a cubic function of three 
variables. 

The equations given in the text [p. 604 above] show that each term in 
any one cycle is a periodic function of the second order of each other term 
in the same cycle. Moreover, it may be shown that each term in any one 
cycle is a periodic function of the thii^d order of every term in either of the 
other two cycles; a sort of relation between the cycles taken per se, and with 
one another, precisely the inverse of what obtains (as already shown) for the 
two cycles of three terms containing the six values of the parameter to the 
biquadratic function of two variables. For as regards that case, it was shown 



608 On Aronhold's Invariants. [60 

in the first part of this paper that the terms in the same cycle are periodic 
functions of the third order of one another, and of the second order of each 
of those not in the same cycle with themselves. 

If we make 

„, = A, ^^^ = B, f^ = C, /-'" =D, 
1 + 2m 1 + 2pm 1 + ^p'm 

pA=A', pB = B', pC=C', pD = D', 

p'A=A", p'-B = B", p-G=C", p-D = D". 

The following table will exhibit all the ternary periods that can be 
formed between the terms of the several cycles : — 

(1) AB'D", (4) BA'C", (7) GA'D", (10) BA'B", 

(2) AG'B", (5) BCD", (8) CB'A", (11) DEC", 

(3) AUG", (6) BD'A", (9) GD'B", (12) DC' A". 

For instance, as an example of the meaning of the table, take line (8), 
namely GB'A". This indicates that A" is formed from B' and G from A" 
in the same way as B' from G, and of course A" from G in the same way 
as G from B' and B' from A" , &c. By means of this table it will easily 
be seen that a term in each of two cycles being given, the term in the third 
which forms with the given two a ternary period may immediately be 



The remarks which I have to add on the nature of the equations for 
finding the parameter m, as well for {x, yf as for {x, y, zf, will be given 
hereafter. 



61. 

ON A REMARKABLE MODIFICATION OF STURM'S THEOREM. 

[Philosophical Magazine, v. (1853), pp. 446 — 456.] 

Let me be allowed to use the term improper continued fraction to 
denote a fraction differing from an ordinary continued fraction, in the 
sole circumstance of the numerators being all negative units instead of 
positive units, as thus : 

The successive convergents of such a fraction as that written above 
will be 

1 g. qsq.-l g^^_ 

?! Mi-1 gsMi-ga-?! 
If we call these respectively 

N, iV, N, . 

we have the general scale of formation 

Moreover, we shall have universally 

N, D,_y - A^,_i A equal to + 1, 

instead of alternating between + 1 and — 1, as is the case in continued 
fractions of the ordinary kind. 

Again, let me be allowed to use the term signaletic series to denote 

a series of disconnected terms, designed to exhibit a certain succession of 

algebraical signs + and — , and to speak of two series being signaletically 

equivalent when the number of continuations of signs and of variations of 

s. 39 



610 On a remarkable Modification of Sturm's Theorem. [61 

signs between the several terms and those that are immediately contiguous 
to them is the same for the two series ; a condition which evidently may 
be satisfied without the order of such changes and continuations being 
identical. I am now able to enunciate the following remarkable theorem 
of signaletic equivalence between two distinct series of terms, each generated 
from the same improper continued fraction. But first I must beg to introduce 
yet another new term in addition to those already employed, namely reverse 
convergents, to denote the convergents generated from a given continued 
fraction by reading the quotients in a reverse order, or if we like so to say, 
the convergents corresponding to the given continued fraction reversed. 

The two forms 

— 1 , and — 1 

■ -1 : _1 

?« qi' 

are obviously reciprocal; and if the two last convergents of either one of 
them be respectively 

-^=^ will serve to generate the other. For the clearer and more simple 

enunciation of the theorem about to be given, it will be better to take as our 

first convergent - , so that 1 will be treated as the denominator of the first 

convergent in every case ; and calling D^ such denominator, we shall always 
understand that D^ = 1. Let now i)„, Di, Da • ■ ■ I^n be the (?i + 1) denominators 
of any improper continued fraction of n quotients, and (To, (Zj, Q^-'-dn 
the corresponding denominator series for the same fraction reversed ; then, 
I say, that these two series are signaletically equivalent. 

I do not here propose to demonstrate this proposition, to which I was 
led unconsciously by researches connected with the theory of elimination, 
vsrhich afford a complete and general but somewhat indirect and circuitous 
proof. Doubtless some simple and direct proof cannot fail ere long to be 
discovered*. For the present I shall content myself with showing a, 'posteriori 
the truth of the theorem for a particular case. Let n = 3. The two series 
which are to be proved to be signaletically equivalent may be written 

1, A, BA-\, GBA-G-A, 

1, C, BC-\, ABC-A-G. 

* See Postscript [p. 616 below]. 



61] On a remarhahle Modification of Sturm's Theorem. 611 

Call these respectively S and (S). In S we may substitute in the third term, 
in place of BA — 1, GA without affecting the signaletic value of the series ; 
for if the second and fourth terms have different signs, the third term may 
be taken anything whatever, since the sequence of the second, third, and 
fourth terms will give one continuation and one change, whatever the middle 
one may be. Suppose, then, that the second and fourth terms have the 
same sign, and let 

CBA-C-A=m''A, 

therefore G {BA - l)=(m?+l)A, 

therefore (BA - 1) AG = (m^ + 1) A\ 

Hence BA —1 and AG will have the same sign; hence S is signaletically 

equivalent to S' , where S' denotes the series 

1, A, GA, GBA-G-A. 

Now, again, if GA is negative, we may put instead of A anything 
whatever, and therefore, if we please, C, without affecting signaletically the 
value of S'. But if GA is positive, A and G will have the same sign, and 
therefore on this supposition also G may be substituted for A. Hence 
always *S' is signaletically equivalent to S", where S" denotes 
1, G, GA, GBA-G-A. 

Again, if G and CBA — G — A have different signs, the value of the 

intermediate term is immaterial ; but if G and GBA —G — A have the same 

sign, let 

CBA-G-A=m^G; 

then A(GB-l) = (l+m-)G, 

and A'{GB-l)={l+m')AG; 

and consequently GB - I and AG have the same sign. In every case, 

therefore, S" is signaletically equivalent to 

1, G, GB-l, ACB-A-G; 
that is S is signaletically equivalent to S', and therefore to >S", and therefore 
to (S), as was to be proved. 

The application of the foregoing theory to Sturm's process for finding 
the number of real roots of an equation is apparent ; for a very little con- 

f'x 
sideration will serve to show, that if we expand ■^j fx being of the nth 

degree in x, algebraically under the form of a continued fraction 

: _\^ 

Qn' 

39—2 



612 On a remarkable Modification of Sturm's Theorem. [61 

where Qj, Q^, Qs---Qn caay be supposed linear functions of x (although, 
in fact, this restriction, as will be hereafter noticed, is unnecessary), the 
denominators of the reverse convergents 

1' Qn' QnQ«-i-l"" Q«e«-i-Qi-&C. ' 

will be signaletically equivalent with the Sturmian series of functions for 
determining the number of real roots of /« within given limits ; in fact, 

will be the Sturmian functions themselves, divided out by the negative 

of the last or constant residue which arises in the application of the process 

of continued division, according to Sturm's rule ; and as we have shown that 

the series of the denominators to the convergents of any continued fraction, 

and the series of the denominators to the convergents of the same fraction 

reversed, are signaletically equivalent, we have this surprisingly new, 

interesting, and suggestive mode of stating Sturm's theorem, namely, the 

denominators to the convergents of the continued fraction which represents 

f'x 

-^ constitute a Rhizoristic series for fx, that is a signaletic series which 

fx 

serves to determine the number of roots of /« comprised within any prescribed 

limits. Moreover, in applying this theorem it is by no means necessary that, 

fx 
in the continued fraction which represents -^ , all or any of the quotients 

should be taken linear functions of x. A very little consideration of the 

principles upon which the demonstration of Sturm's theorem is founded will 

serve to show that the convergent denominators to any continued fraction 

fx 
whatever which represents -^ , whether the quotients be linear or non-linear, 

integral or fractional, or mixed functions of x, and whatever the number 

of quotients, which, it may be observed, cannot be less than, but may be 

made to any extent greater than the exponent of the degree of fx, will 

equally well furnish a Rhizoristic series for fixing the position of the roots, 

fx 
provided only that the last divisor in the process of expanding -^ under the 

form of an improper continued fraction be a constant quantity or any function 
of X incapable of changing its sign. 

Let us, however, for the present confine our attention to the ordinary 
Sturmian form, where all the quotients are linear functions of x. Let these 
quotients be respectively 

a^x + 6i, a^x + 621 o^a* + &3 • • • «„« + &„. 

In order to determine the total number of real and imaginary roots 
oi fx, we must count the loss of continuations of sign in the Rhizoristic 



61] On a remarkable Modification of Sturm's Theorem. 613 

series in passing from a; = + x to « = — oo. When x is infinitely great, it is 
clear that, whether positive or negative, the parts hi,h^...hn may be neglected, 
and only the highest powers of x need be attended to in writing down the 
signaletic series corresponding to these two values of x. Accordingly for 
a; = + 00 the signaletic series becomes 

1, a^x, a^ac^'^, ..., a-^a^-.-anX^, 
and consequently the number of pairs of imaginary roots of fx is the number 
of changes of sign in the series 

1, ai, Oia^, ..., Oia^ ...a„, 
that is, is the number of negative quantities in the series 
«!, ^2, as, ..., an. 

Hence we have the curious and hitherto strangely overlooked theorem, that 
in applying Sturm's process of successive division to fx and /'a;, the number 
of negative coefficients of x in the successive quotients gives the number 
of pairs of imaginary roots oifx; as a corollary, we learn the somewhat curious 
fact that never more than half of these coefficients can be negative ; and in 
general it would appear that the better practical method of applying Sturm's 
theorem would be not to deal with the Residues, which have hitherto been 
the sole things considered, but rather with the linear quotients which have 
been treated as merely incidental to the formation of the Residues. 

To find the value of the Rhizoristic series corresponding to a given value 
of X, the better method would accordingly seem to be to commence with 
finding the arithmetical values of the n quotients 

a^x + 6i, Ota* + &2 • • • «««' + ^a- 
We thus obtain n numbers /uLi, /j^ ... fXn, and have only to form a progression 
according to the well-known law 

where N^ = fi, and in general N, = fjL,N,-.i — N^_^. 

The number of arithmetical operations required by this method (after the 
division part of the process which is common to the two methods has been 
performed) will be* 2n multiplications and 2n additions or subtractions; 
whereas if we deal with the residues directly, the number of multiplications 
will be 

n + (7i-l)+ ... + 1, 

n(n + \) 
that IS n , 

(besides having to raise x to the nth power), and the same number of 

additions. The practical advantage, however, of this method over the old 

[* footnote, p. 622 below.] 



614 On a remarkable Modification of Sturm's Theorem. [61 

method is not quite so great as it may at first sight appear, in consequence 
of the quantities operated with on applying it being larger numbers than 
those which have to be used in the old method. 

If we were to employ, instead of the direct series, 

1, N^, N,N,-l,&;c., 

the signaletically equivalent reverse series 

1, iV„, N'^_^Nn-l,&c., 

the arithmetical difficulty would be much increased in consequence of the 
quotients becoming rapidly more complex as the division proceeds. It were 
much to be desired that some person practically conversant with the application 
of Sturm's method, such as that excellent and experienced mathematician, 
my esteemed friend Professor J. R. Young, would perpend and give his 
opinion upon the relative practical advantages of the two methods of 
substitution ; the one that where the residues are employed, the other that 
where the quotients. 

I am bound to state, that but for a valuable hint furnished to me by my 
friend, that most profound mathematician, M. Hermite, who discovered 
a theorem virtually involving the transformation of Sturm's theorem here 
presented, but founded upon entirely different and less general considerations, 
and in the origin of which hint, as arising out of my own previous speculations 
upon which I was in correspondence with M. Hermite, I may perhaps myself 
claim a share, this theory would probably not have come to light. It is of 
course not confined to Sturm's theorem, which deals only with the special 
case of two functions, whereof one is the first derivative of the other. 

There is a larger theory, to which M. Sturm's is a corollary, which 
contemplates the relations of the roots of any two functions whatever. 
This is what I term the theory of interpositions, upon which I do not 
propose here to enter, but which will be fully developed in a memoir nearly 
completed, and which I shortly propose to present* to the Royal Society, 
wherein will be found combined and flowing into one current various streams 
of thought bearing upon this subject which had previously existed disunited, 
and appearing to follow each a separate course. 

Remark. 

I am not aware that anyone has observed what the effect would be 

of omitting to change the signs of the successive residues in the application 

of Sturm's method, that is, of employing a proper in lieu of an improper 

f'x 
continued fraction to express "^ . 

[* pp. 429—586 above.] 



61] On a remarkable Modification of Sturm's Theorem. 615 

Although easily made out, it is well worthy of being remarked. Suppose 
^ = i 1 



and in general (P being any letter) use P to denote —P. Now we may 
write 

/=Qi</>-Pi, 

<f>= Q2P1- P2, 

Pi = QsRi — Ps, 

Pi=QiP3— Pi, 

Ps=QsPi— Ps, 

Pi= QePs — Pe, 

&C. = &C. 

This gives 

(j> = Q^Pi + P2, 
P1 — Q3P2 + P3, 

P2 = ^4/33 + Pi, 
Ps = QsPi+P5, 

&c. = &c. 

The law evidently being that the quotients change their sign alternately, 
that is in the 2nd, 4th, 6th, &c. places, and remain unaltered in the 1st, 3rd, 
5th, &c. places ; whereas the residues or excesses change their signs in the 
1st and 2nd, 5th and 6th, 9th and 10th, &c., and remain unaltered in the 3rd 
and 4th, 7th and 8th, 11th and 12th, &c. places. The effect is, that if, in 
applying Sturm's method, we omit to change the signs of the remainders, and 
take as our signaletic series 

fsc, / 'so, Pi, P2, -Bs • • • -B»-i, 

Pi, P2, Pg, &c. being the successive unaltered residues, the signaletic index 
corresponding to any value of x instead of being the number of continuations 
in the above series, will become the number of continuations in going from 
a term in an odd place to a term in an even place plus the number of 
variations in going from a term in an odd place to a term in an even place. 

If we adopt the quotient method, the rule will be simply to change the 
sign of the alternate quotients (beginning with the second) in forming the 
signaletic series. 



616 On a remarkable Modification of Sturm's Theorem. [61 

As an artist delights in recalling the particular time and atmospheric 
effects under which he has composed a favourite sketch, so I hope to be 
excused putting upon record that it was in listening to one of the magnificent 
choruses in the 'Israel in Egypt' that, unsought and unsolicited, like a ray 
of light, silently stole into my mind the idea (simple, but previously un- 
perceived) of the equivalence of the Sturmian residues to the denominator 
series formed by the reverse convergents. The idea was just what was 
wanting, — the key-note to the due and perfect evolution of the theory. 



Postscript. 

Immediately after leaving the foregoing matter in the hands of the printer, 
a most simple and complete proof has occurred to me of the theorem left 
undemonstrated in the text [p. 610]. 

Suppose that we have any series of terms u^, u^, u^... m„, where 

Vry = A-^, 'U,^=A^A^—\, Ms = ^1^2-43 — J.1 — J.3, &c. 

and in general 

U, = A, Wj_i — Mj_2, 

then v^, i«2, Us...Un. will be the successive principal coaxal determinants 
of a symmetrical matrix. Thus suppose w = 5 ; if we write down the matrix 



A,, 


1, 


0, 


0, 


0, 


1, 


A,, 


1, 


0, 


0, 


0, 


1, 


A„ 


1, 


0, 


0, 


0, 


1, 


A,, 


1, 


0, 


0, 


0, 


1, 


^5, 



(the mode of formation of which is self-apparent), these successive coaxal 
determinants will be 

1 \A, 



Au 


1 


A, 


1, 





A,, 


1, 


0, 





Au 


1, 


0, 


0, 





1, 


A, 


1, 


A,, 


1 


1, 


A,, 


1, 





1, 


A,, 


1, 


0, 









0, 


1, 


A, 


0, 


1, 


^3, 


1 


0, 


1, 


A„ 


1, 















0, 


0, 


1, 


A, 


0, 
0, 


0, 
0, 


1, 
0, 


A, 
1, 


1 

A 



that is 

1, .0.1, ^3.1^12 — 1> AiA^A^ — Ai — A^, AiA^A^A^ — A-iA^ — A-^Ai — .0.3.0.4-^1, 

A^A^A^^A, - .^1^.2^.6 - A^A^A^ - .43^4^5 - A^A^^ + A^ + A^ + A^. 

It is proper to introduce the unit because it is, in fact, the value of a deter- 
minant of zero places, as I have observed elsewhere. Now I have demon- 



61] On a remarJcable Modification of Sturm's Theorem. 617 

strated directly in this very Magazine (August 1852)*, under cover of the 
umbral notation, that the signaletic value of a regularly ascending series 
of principal coaxal determinants formed from any symmetrical matrix is 
unaffected by any such transposition whatever of the lines and columns 
of the matrix as does not destroy the symmetry about the principal axis. 
Hence, then, beginning from the lower extremity of the axis A^, and reading 
off the ascending series of coaxal minors from that point, we obtain the 
reverse series, 

1, As, ^5^1-1, AsAiAs- A^—As, ^5^1443^.2-^544- ^5^2- ^.3^3+ 1, 

A^AiA^AnAi — A^AiAi — AsA2As — A^A^A-i — A^A^A^ + A^ + A^ + A^. 
Hence we see that the denominators to the convergents of 

^^-T-1 1 

beginning with 1, form a series s'gnaletically equivalent to that similarly 
formed from the fraction 

and the reasoning is of course general, and establishes the theorem in 
question. 

It seems only proper and natural that I should not leave unstated here 
the signaletic properties of the series of numerators to the convergents to 

f'x 

-J- expanded under the form of a continued fraction. 

Let the number of changes of sign in the denominator series for any 
given value a of a; be called D{a), and for the numerator series N{a). 
Then N(a) —N(h) may be equal to, or at most can only differ by a positive 
or negative unit from D (a) — D (6). The relation between these differences 
depends on the nature of the interval between the greater of the two limits 
a and b, and the root of f(cc) next less than that limit, and of the interval 
between the less of the two limits a and b, and the root of ^a; next greater 
than such limit. If a root oif'x is contained in each such interval, 

N {a)-N (b) = I){a)- B{b)^\; 
if a root of / 'x is contained within one interval, but no root within the 
other, 

N(a)-N{b)==I)(a)-D(b); 

if no root of/ 'x is contained within either interval, 

W(a)-N(b) = I)(a)-D(b)-l. . 
[* p. 380 above.] 



618 On a remarkable Modification of Sturm's Theorem. [61 



I may conclude with noticing that the determinantive form of exhibiting 
the successive convergents to an improper continued fraction affords an 
instantaneous demonstration of the equation which connects any two con- 
secutive such convergents as 

namely iV, A-i - i^.-i A = 1- 

For if we construct the matrix, which for greater simplicity I limit to five 

lines and columns, 

A, 1, 0, 0, 

1, B, 1, 0, 

0, 1, C, 1, , (M) 

0, 0, 1, D, 1 

0, 0, 0, 1, E 

and represent umbrally as 

/a,, a^, as, a^, as\ 

Ui, h, h, &4, hj' 
and if, by way of example, we take the fourth and fifth convergents, these 
will be in the umbral notation represented by 



2. ffls) ai\ /«2. «3, «4> fflsN 

i, Is, h) and ^^^' ^^' ^^' ^^' 



/tti, da. o,s, csA /Oi, a^, as, ai, ajV 

\h, h, hs, bj \b„ h, 63, b„ bj 

respectively. Hence 

_ /a,i, as, aA /aa, as, ai, a^, aj\ 
U2, bs, bJ V62, 63, &4, 65, bJ' 
which [p. 252 above] 

_ /Oj, Us, ai, as\ /a^, as, ai, aA _ /a^, a,, ai, cieN fa^, as, ai, aA 

~ Us, 631 bi, bJ V&2, bs, bi, bJ U2, 63, bi, bJ V&2. ^3. h, hJ 

fa^, as, ai, ai\ fa^, a,, ai, a^ 

U2, bs, bi, bJ Us, bs, bi, bJ 

_ /da, as, ai, aA /a^, a^, as, aA 

~ U, b„ bs, bi) "^ \b„ bs, bi, bJ ' 

1, B, 1, 1, 0, 0, =1x1 = 1, 

0, 1, C, 1 

0, 0, 1, D 

0, 0, 0, 1 



that is 



1, 


0, 


0, 


B, 


1, 


0, 


1, 


G, 


1, 


0, 


1, 


n, 1 



61] On a remarTcable Modification of Sturm's Theorem. 619 

as was to be proved. And the demonstration is evidently general in its 
nature. We may treat a proper continued fraction in precisely the same 
manner, substituting throughout \/(— 1) in place of 1 in the generating 
matrix, and we shall thus, by the same process as has been applied to 
improper continued fractions, obtain 

i\r^+, A - iV^. A+. = {V (- 1)}' X {V (- 1)}' 
= (-!>. 

I believe that the introduction of the method of determinants into the 
algorithm of continued fractions cannot fail to have an important bearing 
upon the future treatment and development of the theory of Numbers*. 

* If in the above matrix (M) we write throughout ^^ ( - 1) in place of 1, we have a representation 
of the numerators and denominators of the convergents to a proper continued fraction, and such 
representation gives an immediate and visible proof of the simple and elegant rule (not stated in 
the ordinary treatises on the subject, nor so well known as it deserves to be) for forming any such 
numerators or denominators by means of the principal terms in each ; the rule, I mean, according 
to which the ith denominator may be formed from ?i22?324 • • ■ ?i (?i' ?2"-2i being the successive 
quotients), and the ith numerator from 2223 •••Si,' ^y leaving out from the above products 
respectively any pair or any number of pairs of consecutive quotients as jpjp-i-i. For instance, 
from <ri?2?3?4?6' ''y leaving out q^q^, q^q^, q^qi and q^q^, we obtain 

Miis + SiMs + 2l92?5 + ?l?2'/3 '• 
and by leaving out q^q^ x 3554 , gi^j x q^q^ , q^q^ x q^q^ , we obtain 2g + ?3 + Si ; so that the total 
denominator becomes 

3i?233?4?s + 333435 + 3i3435 + 3i3235 + 3i3233 + 3i + 33 + 35 ; 
and in like manner the numerator of the same convergent is 

(.,111 1 1 

22S33435 ^1 H 1 1 -^ r ' 

I 3233 3334 3436 32333436) 

that is 32333435 + 3435 + 3235 + 3233 + 1- 

The most cursory inspection of the form of the generating matrix will show at once the reason 
of this rule. It may furthermore be observed, that every progression of terms constructed in 
conformity with the equation 

may be represented as an ascending series of principal coaxal determinants to a common matrix. 
Thus if each term in such progression is to be made a linear function of the three preceding 
terms, it wOl be representable by means of the matrix 



A, 


B, 


C", 


0, 





1, 


■^', 


B", 


C", 





0, 


1, 


^", 


B'", 


C" 


0, 


0, 


1, 


A"', 


B"' 


0, 


0, 


0, 


1, 


A" 



indefinitely continued, which gives the terms 

1, A, AA'-B; AA'A"-BA"-AB" + C", . 



62. 



NOTE ON A REMARKABLE MODIFICATION OF STURM'S 
THEOREM, AND ON A NEW RULE FOR FINDING 
SUPERIOR AND INFERIOR LIMITS TO THE ROOTS OF 
AN EQUATION. 

[Philosophical Magazine, vi. (1853), pp. 14 — 20.] 

In my paper [p. 609 above] on this subject in the preceding Number of the 
Magazine, I showed how by means of the quotients a^x + h^, a^x + b^-.-anX+bn, 

f'x 
obtained by throwing-^-;- under the form of a continued fraction, the process 

/^ 
for finding the signaletic index for any given value of x in the series for deter- 
mining the number of real roots of fx within given limits was reduced to 
performing two sets of n multiplications and as many additions or subtractions. 
But by means of a very simple observation, I can now show that the second 
and more laborious set of multiplications may be dispensed with and replaced 
by the simple operation of finding reciprocals, which can be done by mere 
inspection by means of Barlow's or similar tables, which are familiar to all 
computers. If we call the quotients 

OiX + bi, as/v + 62 • • ■ «n* + 6„, 

we must, as explained in the preceding article, find the n numerical values 
/111, Ma ■■• A'n which these quotients assume for any assigned value of x. This 
being done, the signaletic index corresponding to such value of x, that is 
the number of continuations of sign in the signaletic series 

is evidently the number of positive terms in the series 
1, 1 1 1 1 , -1 1 

Ml /Mi , Ma 1 Mn 

Ml M2 Mn-l 

Ml 12 

Ml' 



62] On a remarkable Modification of Sturm's Theorem. 621 

These terms may be found with the utmost facility in succession from 
one another ; for if M, be one of them, the next will be (;ct^+i — M,)~^. Thus, 
then, the necessity for the more operose set of multiplications is done away 
with, and the actual labour of computation reduced much more than 50 per 
cent, below that required by the method indicated in the preceding article 
on the subject. I need hardly add, that the old method of Sturm would 
admit of a similar abbreviation; but in using it we should be subjected 
to the great practical disadvantage of having to begin with the more heavy 
and complicated quotients /u.„, /".n-u &c. instead of /ttj, jj,^, &c., which would 
very greatly enhance the labour of computation. I will conclude by a remark 
of some interest under an algebraical point of view. 

It has been stated that the denominators of the successive convergents to 

: 1 

?i 
are equivalent (to a constant factor pres) with the Sturmian functions, and 
the reader may be curious to know something of the nature of the signaleti- 
cally equivalent series formed by the denominators of the convergents to 
the direct fraction 

i 1 , 

: _1^ 
q» 

These denominators are (abstracting from a constant factor not affecting 
the signs) the Sturmian residues resulting from performing the process 
of common measure between f'x and fiX ; f^x being related in a remark- 
able manner in point of form to fx. Call the roots of /« Oj, cia ... ««; we 
know that/'* is 

2 {{x - a^ (x-as) ...{x- an)}, 

and I am able to state that fiX is (to a constant factor pres) equal to 

S [^(a^, as ■ ■ • dn) {{a> — a^ {x-a^) ...{x- a„))], 
f (a2, Mg ... a,i) denoting the product of the squares of the differences between 
the (w — 1) quantities a^, a^... a„. Accordingly it will be seen that whenever 
X is indefinitely near, whether on the side of excess or defect, to a real root 
of fx, f'x and f-^x will have the same sign ; which serves to show, upon 
an independent and specific algebraical ground, why the two series of residues 

corresponding to ^^ and "^ are (as by a deduction from a general principle 
jx jx 

they have been previously shown to be) rhizoristically equivalent. 



622 On a remarkable Modification of Sturm's Theorem. [62 



Observation. 

In comparing the relative merits of the old and new methods of substi- 
tution for the purposes of Sturm's theorem, the effect of the introduction 
of positive multipliers into the dividends in order to keep all the numerical 
quantities integral ought not to be disregarded. If we call the quotients 
corresponding to this modification of the dividends Q^, Q^, Q^, Qt, &c., and 
the factors thus introduced nii, m^, W3, m4, &c., the true quotients will be 

and it will be found that we may employ as our rhizoristic index either the 
number of continuations of sign in the series 

1, Qu Q^Qi-fn-i, Qs{Q2Qi-'m2)-msQi,Si!ic. 

the law of formation of the successive terms Uo, Ui, U2, &c. being 

or the number of positive signs in the series 



nh 



&c. 



the law of formation of the successive terms Vi, v^, v^, &c. being 

v, = Q. . 

There may therefore, in fact, be in each case (n — 1) more multiplications 
than have been taken account of in the text above. 

If integer numbers be used throughout (so that accordingly the u series 
is that made use of), the total number of multiplications will in general 
be n + 2{n — \)* or 3«- — 2 ; the old method, as previously stated, would 
require ^w(w+l) multiplications; for if we call any one of the Sturmian 
functions 

AaX' + A^x'-^ + Arfic'--^ + ... +A,, 

we shall, using the most abbreviated method of computation, have to calculate 
successively 

xAo + J-i, a; (xAd + A^) + A^, &c., 

* If all the extraneous factors are units, the number of multiplications (Uke that of the 
additions) would be 2«-l, and not 2n, as inadvertently stated in the preceding number of the 

Magazine. 



62] On a remarkable Modification of Sturm's Theorem. 623 

giving rise to i operations (but, it must be admitted, with the practical 
advantage of the use of a constant multiplier) ; and as t may take all 
values from n to 1, the total number of such operations will be \n(n + V). 
When n = 4, 

Consequently (if it be thought necessary to adhere to integers throughout), 
for values of n not exceeding 4, the old method would be probably the 
more expeditious. 



Addendum. 

On a method of finding Superior and Inferior Limits to the real Roots 
of any Algebraical Equation. 

The theory above considered has incidentally led me to the discovery 
of a new and very remarkable method for finding superior and inferior 
limits to the real roots of any algebraical equation. Suppose in general 
that 

:^ = _LJ_J_ 1- 
!>" <h + 1^ + 13 + '" In 

then it is easily seen that 

D = M,M^M^...Mn, 
where 

Mi = 5i, if, = 5, + -, M, = q, + ^^...M„=qn + ^^^. 

In general let any numerical quantity within brackets be used to denote 
its positive numerical value ; so that, for instance, whether 5 = + 3, {q) will 
equally denote + 3. 

And now suppose that neither q^ nor g„, the first or last of the quotients, 
lies between + 1 and — 1, and that no one of the intermediate quotients 
§2, qs-.- qn-i lies between + 2 and — 2 ; so that, in other words, 

(2O > 1, {q,) > 2, (23) > 2 . . . (?„_0 > 2, (5„) > 1 ; 

then, I say, that ilfj, M^, M^ ... Mn will have the same signs as qi, q^, q^.-.qn 
respectively ; for 

M, = q„ 

therefore (ilfi) > 1 ; 

but M, = q, + ^, 



624 On a remarTcahle Modification of Sturm's Theorem. [62 

therefore {M^) = (g^) + [-^ j > 2 ± 1 ; 

therefore 

ilfj has the same sign as q^, and also (M^) > 1 ; 

therefore in like manner, 

(Ms) has the same sign as ^j, and also (M^) > 1 ; 

therefore in like manner, 

(if 4) has the same sign as ^4, and also (if 4) > 1 ; 

and so on until we come to if n-i, and we shall find 

if„_i of the same sign as qn^i, and also (if^-i) > 1. 

Finally, 

Mn = qn±J-, 
"•' »— 1 



where (qn)> 1 and (-jrf — j < 1, therefore 



Mn has the same sign as qn ; 

but we cannot say (nor is there any occasion to say) that (if„) > 1 ; therefore 

D = Mj^M^Ms . . . Mn has the same sign as qiq^qs ■■■qn- 

Now let fx be any given function of x of the nth degree, and <f)x any 
assumed function whatever of x of the (n — l)th degree, and let 

^_ J_ J_ J_ 2. 

/« ~g'i + ?2 + g'3 + '"?»' 

where ^i, q^, q^-.-qn are now supposed to be linear functions of «, which, 
except for special relations between / and <p, will always exist, and can be 
found by the ordinary process of successive division. 

Write down the n pairs of equations, 

^1 = 31 + 1 = 0, ^2 = ^2 + 2 = 0, M3 = g'3 + 2 = 0...M„ = g'„+l = 0, 
m'i = 5'i — 1=0, M'2 = g'2— 2 = 0, M's = g3 — 2 =0 ... M'„ = g'M— 1 = 0. 

If the greatest of the values of x determined from these 2?i equations be 
called L, and the least of these values be called A, it may easily be made out 
that between + ao and L, each of the quantities qi, q^, qs ■■■ qn will remain 
unaltered in sign ; and between — 00 and A also the same invariability of 
sign obtains ; and, moreover, between + 00 and L, and between A and — 00 , 
(31). (^a) • • ■ (^n-i). (qn) will be respectively greater than 1, 2... 2, 1. Con- 
sequently, by virtue of the preceding theorem, between + 00 and L, and 
between A and — 00 , D will always retain the same sign as giga^s ••• S'n.j 



62] On a remarkable Modification of Shirm's Theorem. 625 

and therefore no root of /a; will be contained within either such interval. 
And hence /a;, which is manifestly identical with D (the denominator of the 
continued fraction last above written), affected with a certain constant factor, 
will retain an invariable sign within each such interval respectively. Hence, 
then, the following rule. 

Calling 5'i, q^, 53 ... g„ respectively 

a^x — bi, UnX — 62, ctaa; — tj ... a„a; — bn, 

if we form the 2n quantities 

b,±l 6; + 2 63 + 2 6„_i + 2 bn±l 

the greatest of these will be a superior limit, and the least of them an inferior 
limit to the roots oifx. 

The values of these fractions will depend upon the form of the assumed 
subsidiary function <^. Hence, then, arises a most curious question for 
future discussion — to wit, to discover whether in any case the subsidiary 
function can be so assumed as that the superior limit can be brought to 
coincide with the greatest, or the inferior limit with the least real root, 
supposing that there are any real roots. I believe that it will be found that 
this is always impossible to be done. Then, again, if all the roots are 
imaginary, can inconsistent limits (evincing this imaginariness) be obtained 
by giving different forms to the subsidiary function, which would be the case 
if we could find that the superior limit brought out by one form were less 
than the inferior limit brought out by another, or the inferior limit brought 
out by one form greater than the superior brought out by another ? If, as I 
suspect, this also can never be done, then the general question remains to 
determine for all cases the form to be given to the subsidiary function, which 
will make the interval between either limit and its nearest root, or between 
the two limits themselves, a minimum. Thus, it appears to me, a fine field 
of research is thrown open to those who are interested in the theory of maxima 
minimorum, and minima maximorum, and one likely to lead to unexpected 
and important discoveries [cf p. .533 above, and the Author's footnote, p. 495]. 

It may be asked how is the above rule to be applied if any of the leading 
coefiicients in (j}X, or of the successive residues of fx and (px vanish ; in 
which case, instead of the coefficients being linear, some of them will be, as in 
fact all might be, polynomial functions of x. The rule, it may be proved, 
will still subsist. 

Equating the first and last quotients each of them to + 1 and to — 1, 
and the intermediate ones to + 2 and to — 2, the greatest root of all the 
equations so formed continues to be a superior, and the least root an inferior 
s. 40 



626 On a remarkable Modification of Sturm's Theorem. [62 

liiuit to the roots of/a;. Nor is it ever necessary, even in these special cases, 
actually to solve any of these equations ; for evidently it will be sufficient to 
find a superior limit and an inferior limit to each of them, and adopt the 
greatest of the superior and the least of the inferior limits as the superior 
and inferior limits to the roots of the given equation. Thus, then, we should 
have to repeat upon the quotients increased and diminished by 1 or 2 (as the 
case may be), the same process as is supposed to be originally applied to fx, 
and thus by a continued process of trituration (since every new function 
so to be operated upon is of a lower degree than the original function) we 
must finally descend to linear equations exclusively. 

It is interesting thus to see that there are no failing cases in the 
application of the rule, and that a solution of equations of a higher degree 
than the first is never necessary. But as a matter of fact, the chances 
are infinitely improbable (if ^x is chosen at random), of any of the quotients 
after the first ceasing to be linear ; and the first is of course linear, provided 
that the degree of </>« is taken only one unit below that oifx. 

In working with Sturm's theorem, a system of quotients is supplied ready 
to hand ; and these quotients, by virtue of the rule given above, may be used 
to assign a superior and inferior limit in the first instance, before setting 
about to determine the distribution of the roots between these limits by aid 
either of these same quotients or of the residues. For the change of sign 
of the residues required by the Sturmian process will only affect the signs, 
and not the forms of the quotients ; but in the application of the above rule 
for finding the limits, the sign of any quotient is evidently immaterial. 



63. 



ON THE NEW RULE FOR FINDING SUPERIOR AND INFERIOR 
LIMITS TO THE REAL ROOTS OF ANY ALGEBRAICAL 
EQUATION. 

[Philosophical Magazine, vi. (1853), pp. 138 — 140.] 

The lemma accessory to the demonstration of the rule for finding limits 
to the roots of an equation, given in the addendum [p. 623 above] to my 
paper in the Magazine for this month, admits of two successive and large 
steps of generalization, in which the scope of the principal theorem will 
participate in an equal degree. 

1. Whatever the signs may he oi q^, q^, qs ... qr, the denominator of the 
continued fraction 

J_ J^ 1 J. 

qi + q2+ qs'" qr 

will have the same sign as qiqoqz... qr, provided that 
[9i]>fh> [g'2]>/^2 + — , [q3]>fis + — ... 

fh. f^2 

r n 1 r n 1 

H'r-2 P»-i 

where /tt,, /j^ ... fi,—! signify any positive quantities whatsoever; in the 
particular case where /j^ = fi^ = fji^= ... = fi^_^ = 1, we fall back upon the lemma 
as originally stated. 

2. But the lemma admits of another modification, which will in general 
impose far less stringent limits upon the arithmetical values of the series 
of q's. 

Let all the possible sequences of q's be taken which present only variations 
of sign ; for example if the entire series be q^, q^, q^, qt, and the corresponding 

algebraical signs are -\ h, we shall have the two sequences qu q^', S's, ?4. 

If the entire series be q^, q^, q^... q^^, and the signs be 

1-- + + + +- + + + + -, 

40—2 



628 Ofi the New Rule for finding Superior and Inferior [63 

then the sequences to be taken will be 

q^, qu qs, q^; q^, qw, qn, qu, q^, 

and so in general. 

Suppose, now, that gp+i , jp+a • • • qp+i are the terms of any one such sequence. 
Then, provided that 

[qp+i] > /ii. [g'p+2] > /x, + — . . . g'p+f-i > /ii_i + , 

^ /^i— 2 

and q^+i > — , 

(it being understood that the values of /ij, /j,^ ... fii_i are perfectly arbitrary, 
except being subject to the condition of being all positive, and that there 
are as many distinct and independent systems of such values as there are 
sequences of variations of sign), it will continue to be true (and capable of 
being demonstrated to be so by precisely the same reasoning as was applied 
to the demonstration of the lemma in its original form) that the denominator 

of — ... — will have the same sign as the product qiq2qs... qr. It will 

be observed that, as regards the residual quotients not comprised in any 
sequence, their values are absolutely unaffected by any condition whatever. 
As a direct consequence from this lemma, we derive the following greatly 
improved Theorem for the discovery of the limits. 

Let, as before, fx = be any given algebraical equation ; ^x any 
assumed arbitrary function of x of an inferior degree to that of fx; 
and let 

^_ _1 1 1_ 1^ 

fx~ X,^X, + X,+ -Xr' 

let the leading coefficients of X^, X^, X^-.. X^ be qi, q^, qz ■•■qr, and let 
this latter series be divided into sequences of variations and residual terms 
not comprised in any such sequence, as explained above. Let the X's 
corresponding to the residual terms be called 

and let the successive sets of X's corresponding to the sequences be called 
respectively 

F/, V- ... V',, 

y", F"...F'>, 



(F),(F)...(F<p,). 



63] Limits to the real Roots of any Algebraical Equation. 629 

And let 

x(rr-cr)(F,"--c,=)...(F/-c/) 

&c. &c. 

X {(FO^-(cOl{(F.r-(c,)=) ... {(F„,)=-(c,p,)^}, 
where, in general, any system of values 

Ci , C2 , C3 ... Cp — 1 , Cp , 

represents 

1 11 

/ii, /.,+ -.../ip_i + ^— , — . 
Ml A'p— 2 /^p-i 

Then the largest root of X = is a superior limit, and the smallest root of 
X = is an inferior limit to the real roots of /« = ; and if X = has no real 
roots, neither will fx = have any. For the complete demonstration and 
some further developments of this theorem see the forthcoming number of 
Terquem's Nouvelles Annales for the present month*. 

[* p. 423 and p. 424 above.] 



64 

NOTE ON THE NEW RULE OF LIMITS. 
[Philosophical Magazine, vi. (1853), pp. 210 — 213.] 

It may appear like harping too long on the same string to add any 
further remarks on the rule relating to so simple and elementary a matter 
as that of assigning limits to the roots of a given algebraical equation ; 
but it will be remembered that some of the greatest masters of analysis, 
including the honoured names of Newton and Cauchy, have not disdained 
to treat, and to give to the world their comparatively imperfect results 
on this very subject. I hope, therefore, to stand excused of any undue 
egotism in adding some observations which may tend to present, under a 
clearer aspect and more finished form, the new and beautifully flexible rule 
laid before the readers of this Magazine in the two preceding Numbers. 

Firstly, I observe that any succession of signs may be considered as 
made up of, and decomposable into, sequences of changes exclusively, if we 
agree to consider, where necessary, a single isolated sign + or — as a sequence 

of zero changes. Thus, for instance, H 1- + + H h + -\ 1 may be 

treated as made up of the variation sequences 

+ -, - + , +, +, +- + , +, +-+-, -*. 

Secondly, I observe that if Xj, X^...Xi be all linear functions of x, 
and the signs of the coefficients of x in these functions constitute a single 
unbroken series of variations, the denominator of the continued fraction 

1 1 ]_ 1_ 

x, + x;+x,+-"Xi 

(reduced to the form of an ordinary algebraical fraction) will have all its 
roots real. 

* The rule is, that the given series of signs is to be separated into distinct sequences of 
variations, so that the final term of one sequence and the initial term of the next shall form a 
continuation, that is we must have variation sequences connected together by continuations at 
their joinings. 



64] Note on the New Rule of Limits. 631 

Thirdly, suppose, for greater simplicity, that ^x is of one degree in x 
lower than fx, and that by the ordinary process of common measure we 
obtain 

05 1^ Jl ]_ 1 

fx~X, + X^ + X, + -Xn' 
where Xj, X^, X^ ... Xn are all of them linear functions of x. 

Let Xi, Xj ... Xn be divided into distinct and unblending sequences, 

X1X2 ... Xi, Xi+iXi+2 ■■■ Xi', Xi'+i ... Xi", ..., X(i)+iX(i)+2 ■•• Xn', 

so that in each sequence the signs of the coefficients of x present a single 

unbroken series of variations, which by virtue of observation (1), may be 

considered to be always capable of being done, and let 

(^_J^ 1 ]_ J^ 

/i« Xi + X2 + X3 + '" Xi' 

<f>^_ 1 1 1_ 

f^x Xi+i + Xj+2 Xi' ' 



(<j})x_ 1 J^. 

(/)«~Z,i,+i+ Xn' 

then, according to observation (2), the equations 

f,x = 0, f,x=0...if)x = O, 

have each of them all their roots real ; and the observation now to be made 
is, that the highest of the highest roots and the lowest of the lowest roots 
of these equations furnish respectively a superior and infeiior limit to the 
roots oifx = 0*. 

* This theorem may be more concisely stated as follows : — " If U with any subscript be 
understood to mean a linear function of x in which the sign of the coefEcient of x is constant, 
then the finite roots of the equation 

J. 1 1_ ^_ _1 1^ ]_ 1 1 J__ 

U,~ U,- U,~- Ui+ Ui+,- Ui+,- - Ui'+ - C/m+1- i7,te- - U„-°° 
lie between the greatest and least finite roots of the equations 
_J. 1_ J^_ 

1 1 J^_ 



1 1 J__ „ 

Ufa- Ua+,--u,-'"- 



The theorem under this form suggests a much more general one relating to para-symmetrical 
determinants, that is determinants partly normal and partly gauche, which will be given hereafter; 
one example among the many confirming the importance of the view first stated in this Magazine 
by the author of this paper, whereby continued fractions are incorporated with the doctrine of 
determinants. 



632 



Note on the New Rule of Limits. 



[64 



N.B. The single root of any one or more of these which may be of the 
first degree in x is to be treated, in applying the preceding observation, 
as being at the same time the highest and the lowest root of such equation 
or equations. 

Fourthly and lastly, the problem of assigning limits to the roots Ciifx = 
reduces itself to that of finding limits to 

/^^ = 0, /,a^ = 0. ..(/)«;= 0; 

for the greatest and least of these collectively will evidently, d fortiori, 
by virtue of the preceding observation, be limits to the roots of fx = 0. Of 
any such of these as are linear, the root or roots themselves may be treated 
as known ; leaving these out of consideration, the functional part of any 
other of them, such as fx, is the denominator of a continued fraction of the 
form 

1 1 1 1 

(«!» + 6i) + (a^ + 62) + (a^oa + 63) + " (fl-i^ + ^i) ' 

in which a^, a^, a,... ai present a single sequence of variations of sign, and 
the limits to the roots of/i* = may be found as follows. 

Form the two systems of equations (in which /ij, yOa ... /tij_i are numerical 
quantities having all the same algebraical sign, but are otherwise arbitrary 
and independent). 



Or^x + bi = 

diPC + 62 = 

a^x + 63 = 


Ml 
-M2- 

M3 + 


1 

Ml 
1 
M2 


«!« + &i = 

as* + 63 = 


-Ml 

1 

1 
-^'- M. 


ai_-,x + hi--, = (- 
aix + 6i = 


-y->i-i+ 


(_)i-. 

Mi-2 

(_y-i 
Mi-i 


ai-,x + bi-, = (- 
ttiX + hi = 


Mi— 2 

Mi-i 



then (supposing /^i to have the same sign as a-^ the highest of the values 
of X obtained from the first system, and the lowest of the values of x found 
from the second system of these equations, will be a superior and inferior 
limit respectively to the roots of fx — ; and so for all the rest of the 
equations 

Mx) = 0,f,{x) = 0...(f)x=0, 

excluding those of the first degree. 

It will be seen that the theorems contained in the observations (3) and 
(4) combined (which presuppose the statements made in observations (1) 



64] Note on the New Rule of Limits. 633 

and (2)), contain between them the theorem given in the last Number of the 
Magazine [p. 627 above], but rendered in one or two particulars more simple 
and precise, and, as it were, reduced to its lowest terms. In the whole 
course of my experience I never remember a theory which has undergone so 
many successive transformations in my mind as this very simple one, since 
the day when I first unexpectedly discovered the germ of it in results 
obtained for quite a different purpose. In fact, it never entered into my 
thoughts that in so beaten a track, and in so hackneyed a subject as that 
of finding numerical limits to the roots of an equation, there was left any- 
thing to be discovered ; and my sole merit, if any, in bringing the new rule to 
light, consists in having been able to detect the presence and appreciate the 
value of a truth which fortune or providence had put into my hands. 



65. 



THE ALGEBRAICAL THEORY OF THE SECULAR-INEQUALITY 
DETERMINANTIVE EQUATION GENERALIZED. 

[Philosophical Magazine, vi. (1853), pp. 214 — 216.] 

Art. 1. Let 



Vax + a, hx + /3, dx + S 
'ax + a, bx + ^1 

"^ hx + l3, ca; + 7, ex + e 

dx + S, ex + e, fx + <^ 



rax + a, 6a; + pTi 
Xi = aa; + a, Z2 = , X3 = 

Lhx-'r^, cx + 'yA 



&c.. 



and let the first coefficients of X-^, X^,, X^, &c. have all the same sign ; then 
I say that the roots of any such function as X; will be all real, and will lie 
respectively in the intervals comprised between + 00 , the successive descending 
roots of Xf_i and —00. When a = l, c = l, /=1, &c., and & = 0, d = 0, 
e = 0, &c., Xi = becomes the well-known secular-inequalily equation. 

Demonstration. For greater simplicity, let all the first coefficients be 
taken positive, and suppose the theorem proved up to i, it will be true 
for i + 1. For by a well-known property of symmetrical determinants, when 
Xi = 0, Xi_i and Xj+i will have contrary signs. Let the roots of Xi_i be 

hi, A2 ••■ ^i-l> 

and the roots of Xi, 

When x = ki, which is greater than h^, the greatest root of Xi_i will be 
positive ; when x = ^'2, which lies between the first and second roots of 
Xi_i, Xi_i will be negative ; and so on, Xj_i alternately becoming positive 
and negative as we pass from root to root of X;. 

Hence Xi^^, which is positive when x= cc , becomes negative when x = ki, 
positive again when x = k^, and so alternately ; being finally, when x = ki, 
positive or negative, and when a; = — 00 , negative or positive, according as 
i is even or odd. Hence X^+i, which changes sign i+1 times between + 00 
and — 00 , must have all its roots real, and lying severally in the intervals 
included between -f 00 , the successive roots of Xj and — 00 . Hence if the 
theorem be true for i — 1 and i, it is true for all numbers above i ; but if 

we take 

, [ax -fa, bx + B'\ 
ax + cL and 

Lhx + /3, ex + yj 



65] Secular-inequality Determinantive Equation. 635 

the latter is (ax + a) (ex + 7) — (hx + /3)-, which is positive for a; = 00 , negative 
for ax + a = 0, and positive for a; = — 00 . Hence the theorem is true for X^ 
and Xo, and therefore universally. 

In the above demonstration it was supposed that the leading coefficients 
are all positive ; but the demonstration will be precisely the same, mutatis 
mutandis, if they are all negative. 

Art. 2. And "inuch more generally it may be shown, in like manner, that 
if the successions of signs, in the series consisting of the sign + followed by 
the signs of the principal coefficients in Xj , X2 . . . X^+n, consist of m variations 
and n continuations, the number of real roots of the equation X,„+,i = will 
be at least as great as the positive value of the difference between m and n. 
This theorem, moreover, remains true if X,, X„, X3, &c. be formed from 
a symmetrical matrix, in which the terms, instead of being linear functions 
of X, are any odd-degreed rational integral functions of x, or fractional 
functions of which the numerators (when rendered prime to their denomi- 
nators) are odd-degreed functions of x. My friend M. Borchardt, who has so 
beautifully effected the decomposition of my formulae for the Sturmian 
criteria of reality into the sums of squares for the secular-inequality form of 
the equation, may now, if he pleases, tax his ingenuity to effect a similar 
decomposition for the general case supposed in Art. 1*. 

Art. 3. It is obvious that, in applying the theorem contained in 
Arts. 1 and 2, it is indifferent whether we look to the signs of the successive 

determinants a; , ' ; &c., or to those of a ; ' ; &c. ; or, more generally, 

\b, cV 1/3, 7 I 

to those of a-l-«^; , ^^' /, ; &c., 6 being any arbitrary but real 

quantity. Conversely we obtain the remarkable theorem, that wlien any 
homogeneous quadratic function, whose coefficients are linear functions of 6, 

* So, too, my own more simple method for proving the omni-reality of the roots of the 
seoulai--ineqiiality equation, August 1852, [p. 364 above], ought to be capable of being extended to 
the general form in Art. 1, that is we ought to be able to prove that the equation whose roots are 
the squares of the roots of Xj=0 will have all its coefficients alternately negative and positive. 
If we take for example i = 2, the equation to the squares of the roots becomes 

{ac -bY-x'-'- {{ay + ca-2b^)- + 2(b^-ac) [ay -p^)}x + {ay -p^)^=0; 
and we have to prove that the coefficient of - x in this equation is essentially positive when 
ac-b^ is positive: this may be shown by various modes of decomposition; amongst others, 
by writing the coefficient in question under the form 

■4 {{c"a + yb'--2bcp)- + y- {ac - b^- + 2 {by- c^)- {ac - b^)}. 

In general, if L is essentially positive when L^, ij ... ij are positive, then, discarding all 
artifices of calculation, this must be capable of being proved by virtue of an identity of the 
form 

L = Xm^ + Smj- Lj + 2m„- L^+ ... + 'Zmf Lj . 



636 Secular-inequality Determinantive Equation. [65 

is linearly converted hy real substitutions into a sum of positive and negative 
squares, the greatest difference for any value of 6 between the number of 
positive and the number of negative squares has for its limit the number of 
real roots of 6 in the Discriminant {otherwise called the Determinant) of the 
given function. The theorem actually demonstrated above teaches only this 
much, namely that the maximum difference in number between the two 
species of squares (which depends only on the value given to 6) cannot 
exceed the number of real roots in the discriminant; it admits, however, 
of an easy proof that this maximum difference is equal to the number of real 
roots, so that the one number is, in the strict sense of the word, an exact 
limit to the other. 

Art. 4. I was led to the theorem, as given in Art. 1, by having to 
consider the following curious and important question. 

" Given i linear functions of x, say Xj, X^ ... Xi, to find the i—1 positive 
quantities, say /j.^, fi„... Mi-i, which shall give the least value to the greatest root, 
or the greatest value to the least root, of the equation 

ix,-^) \x, - (^ +1)] [x, - (.3+^-3] ... [x, - (^-L)] = 0." 

The theorem in Art. 1 enables me easily to demonstrate, that if we take 
X-[, X2, X3' . . . X( identical with 

VI. Xi, ^J\.X^...^J\.Xi, 

the sign of the square root being selected iu each case so that the coefficients 
of X in X/, Xi ... Xi shall have all the same sign, then the least value of 
the greatest root, and the greatest value of the least root, of the given 
equation will be respectively the greatest and least finite roots of the 
equation 

X'- 1 1 J--0*- 

^' xT^x;'-x{-^ ' 

the two systems of values of /ij, /n„ ... ^;_i required being the two systems 
of values of 

'V'/xr/ ^ -Tr/ -L -L XT-/ -^ ■'- ■*- 

-A 1 ) -^2 ~ ^pw , .^3 xrT ~=v^i ■■■ -^ i-\~ V ' ^V^> • • • "V' ' 

-Ai A.O — ^i -a. i_2 — -a. i_3 ^1 

corresponding respectively to these two values of x. 

And it is by means of this solution that the statement of the rule for 
finding the superior and inferior limits to the real roots of an algebraical 
equation made in the last August Number of the Magazine, is capable of 
being converted into the statement contained in the third observation on 
the same rule in the present Number [p. 631 above]. 

* The finite roots of this are the same as those of 

111 



66. 

ON THE EXPLICIT VALUES OF STURM'S QUOTIENTS. 

[Philosophical Magazine, vi. (1853), pp. 293 — 296.] 

By Sturm's quotients is of course meant to be understood the quotients 

which result from applying the process for the discovery of the greatest 

common measure between fx (an algebraical function of the nth degree 

in OS, and whose first coefficient is unity) and f'x its first derivative, as in 

f'x 
Sturm's theorem ; or which is the same thing in effect, supposing "^ 

to be represented by 

1 1 1_ 1^ 

(where Qi, Qa--- Qn are all linear functions of x), the quotients in question 
are Qi, Q^-^-Qn- Before proceeding to discuss these quotients, it will be 
well to state the form under which the other quantities which appear in the 
course of the application of the Sturmian process admit of being represented. 
First, then, it will be remembered that the residues with the signs changed 
are all of the form 

Ei = Mil {^{h, ,h...hi){x- hi^,) {x - hi+,) ...(x- hn)], 

where ^(h^, h^.-.hi) indicates the squared differences between every two 
of the quantities h^, h^ ... hi, and h^, h^...hn are supposed to be the n roots 
oifx; and where, using ^i to denote Sf (Ai, ^2 ••• h), with the convention that 
?o = Ij ^1 = n, and understanding by (i), ^ |1 + {—)'}, 



Mi 



yi. f2. in 

b 1—2 b 1—4 • • • t 



(i)+l 



Here it will be observed that the only quantities appearing are the factors 
and the differences of the roots of fx ; and since these latter are the same 
as the differences between the corresponding factors, for 

(x— h)- {x- h') = h' — h, 



638 On the explicit Values of Sturm's Quotients. [66 

the entire quantity which expresses any residue Ri may be considered as 
a function of the factors of fx exclusively. 

Again, if we solve the syzygetic equation 

Nifx+Difx = Ru 

I have published many years ago in this Magazine the value of Di, and subse- 
quently in a paper read before the Royal Society on the 16th of June last [p. 429 
above] the value of i\^j,both which values are also functions of the factors oi fx 

N- 
exclusively, -rr, it is easily seen, represents the successive convergents to 

. fx . 
the continued fraction by which -^ is supposed to be expressed, and jB; (to 

a constant factor pres) is the denominator of the reverse convergents of the 
same continued fraction. To the completion of this part of the theory it 
evidently therefore becomes necessary to express the quotients Qi, Q^, Q3... 
Qn-i, Qn (of which the first (n — l) are those which appear in Sturm's process, 
and the last is simply the penultimate Sturmian residue divided by the 
ultimate residue) under a similar form, that is as functions exclusively 
of the factors of fx, or, which comes to the same thing, of the factors and the 
differences of the roots. Guided by an instinctive sense of the beautiful and 
fittincf, in a happy moment I have succeeded in grasping this much wished 
for representation, with which I propose now and for ever to take my farewell 
of this long and deeply excogitated theorem. 

If we write [cf p. 499 above, and the Author's footnote, p. 495] 
iJ._j = Mi_, {4i_i«"-'+i - 5i_ia,-"-* + &c.}, 

and 

Ri = Mi {AiX'"-' - BiX'"-'-' + &c.}, 
we have 

Ai_, = Sr (Ai, /12 • • • hi^i), Bi_, = X{hi + hi^, +...+K) ^Qh, h... hi_,), 

Ai = ^^ Qh, h ■ • ■ k), Bi=% {hi+^ + hi+, + ... + K) ^{h„ h... hi), 

and the ith quotient is evidently 

Mi-y Aj^.AjX + (Aj^^^Bj - AiBi_,) 
Mi Af ' 

and this is the quantity (unpromising enough in aspect) to be transformed 
in the manner prescribed. 

Mi-i, Mi, and Ai are already given under that form, and I find that, 

putting 

Ti = Ai_,AiX + {Ai^.Bi - AiBi_,), 

Ti may be represented by the double sum 

t {[2 {? {he,, he, ■ ■ ■ he, J Oh - he,) {k - he,) ... (A, - he,_M (x - k)}. 



66] On the explicit Values of Sturm's Quotients. 639 

This of course implies the truth of the identity 

in itself a truly remarkable equation, which it will be seen is of 2(i—iy 
dimensions in respect of the roots*. 

When i = 1, 

T, = lix- h,) ; 
and when i=2, 

that is =2 {[(n - 1) /^ -(k^ + h,+ ...+ h„)Y {x - h,)}. 

When i = n, Tn becomes 

2 {r (^, ^ . . . K) X i; (lu, h... K) {x - K)] =^^n + '^{^(h,hs...h„)(w- h)}, 

as it evidently ought to do. Substituting for 2'j_i, Tt and Ai, their values, 
we have as the complete general expression of the ith Sturmian quotient 
the following expression, in which, agreeable to a notation which I have 
previously used and explained. 



^ means (\-Jie^)(h,-he^)...(h,-hg._^), 

It ought not to be passed over in silence, that if we write 

J__J_J_ 1 _ N'i(x) 
Q.-Q.-Q,-"'Qi D.i{xy 

and if we suppose Niix) and Di(x) to be expressed integrally, and to be 
algebraically prime to one another, then 



namelv 



t 2 



Oui. 



* Thus if !i = 4 and i = 2 
and we have 

4 { (fti - h,r- +(h,- h,r- + (/ii - h,r- + (h^ - h^y^ + {i,, - h,y + (h, - h,r-} 

= {3Ai - h„ - A3 - 7(4)= + (37(2 - /»! - fts - 714)= + (37i3 - ;si - h„ - h^Y + (^K - K - h - ''3)". 

and so in general ti-itit which is the product of two sums of variable numbers of squares, 
is expressible rationally as the sum of a constant number (n) of squares for all values of i. 
+ (i) denotes J{(-1)' + 1}. 



640 On the explicit Values of Sturm's Quotients. 



[66 



Hence Qi is contained as a factor in 

(Bi_^Jhy (x - h) + {Di_Xf (oo-h)...+ {Di_,Ky {x - K). 

It may be observed also, that for all values of i between 1 and n 
inclusively, 

DX + Dih + Dih + . . . + DiK = 0, 
and also that the determinant 

1, 1, 1, ... 1 

{BAY, {DXy, (DAT ...{D.Ky 

(DAY, {i>A)\ {D^hy ...{B^Ky 



(B^-AiY, {Bn-Af, {Bn-Ar...{Bn-.hnr 

is always zero [cf p. 502 above]. To complete the theory, I subjoin the value 

f'x 
of Ni, the simplified numerator of the tth convergent to -^ , expressed 

as an improper continued fraction. 

Let the sum of the products oi x — h, x—k...x — l combined i and i 
together be denoted by /Sj (A, k ... I), and the sum of the ith powers of the 
same by ai{h, k ... I), then Ni is equal to 

^?(K> K---K) X Wi-i(K' K ■■■he,)-cri-.,(he„ he, ... he) S^ {hg,^,... he„) 
+ o-i-3 (he^, he^...he)So. (/(«;+, . . . AeJ + &c. 
...±(i + l)S^,ihe,,,...heJ]. 

The anomaly of the last term being of the form (1 + o-o) St-^ (for of course 
o-„ = i), instead of being o-o<St_i, is not a little remarkable. 

Of the four sets of Sturmian quantities, namely the residues, the quotients, 

f'x 
and the denominators and numerators of the convergents to '^- , it will have 

/^ 
been seen that the first and third are expressible in terms of the roots and 
factors by single summations of equal simplicity, the second and fourth by 
double summations, whereof that which corresponds to the numerators is 
much the more complicated of the two. 



67. 



ON A FUNDAMENTAL RULE IN THE ALGORITHM OF 
CONTINUED FRACTIONS. 

[Philosophical Magazine, VI. (1853), pp. 297—299.] 

Let &c. be any continued fraction, and let the successive 

Oi + aa + (Is + 

111 N N 

convergents -, , &c. be called -7^, tt. &c-j and let Di be denoted 

° cti Oi + as A -^2 

by (Oi, a^...ai)*, then the following identity obtains which I regard as the 

fundamental theorem in the theory of continued fractions, but which I have 

never seen stated in any work where this subject is treated [cf pp. 530, 618 

above]. 

Theorem. 

(Cli . . . am) X (tfm+i • • • 0,m+n) + (<^ ■ • • C^m-i) ^ (C^m+a • • • <^m+n) 
= (ffii ... Clm , am+i ■ ■ • am+n)- 

Corollary 1. 

(Oj, as . . . a™) X (a^, as . . . a,^+i) - {a.^, as... a^) x (oi, aj . . . a^+i) = (-)'" 1. 
This is the well-known theorem 

which, however, is only a case of a much more general theorem easily deduced 
from the fundamental theorem given above. In fact, we may derive im- 
mediately from the latter, the equation 

(oi, tta ... a^) X (ttj, as ... a„,+f)- (a2, as ... a^) x (aj, aj ... a^+i) 

= (-)'" (a^+i, a,„+j_i ... to i- 1 terms). 

* It is essential to notice that (a^, a„... af) = (ai, ai_i...a^). 
S. 41 



642 On a Fundamental Rule in [67 

Hence 

Bm-iNm - Ah.-^«!-4 = (-)™ (a,„a,«-iam-2 + am + am-2). 
&c. &c. 

Corollary 2. 

(«! ... ttp, ffip+i ... a.p+/)(ai ... dp, ttp+i ... ap+i) 

— (cti ... ap, ttp+i . . . (Xp+j,) (Oi . . . Kp, a.p+1 . . . ap^k) 

= (-)" {(^P+i • • • «P+/) (ttp+i ■ • • C^p+fc) - (Ctp+i . . . Kp+j) (ctp+i . . . ttp+ft)}. 

Suh-corollary. If all the several quantities Oi, aj, ctg ... are equal to one 
another, the quantity -D/Di — DgBj^ is constant in magnitude, but alternating 
in sign, so long as the differences of the indices /, g, h, k are constant; 
and as an easy deduction from this sub-corollary, if 

Z,+, = aT,,-bT^, 

be the characteristic equation of a recurrent series, and if /+ k = g + h, 

'V f T T 

— — -j^^- — will be constant ; and as a particular case of this deduction 

b^ 
from the sub-corollary to the second corollary of the fundamental theorem, 
we have 

■'■ n -*■ n—1 J- n+i 



6» 
that is 

rWi - aT„Tn+, + bT\ 
6» 



= a constant, 



= a constant, 



which is Euler's theorem. See Terquem's Nouvelles Annales, Vol. x. p. 357, 
and November 1852. 

I was led up to a knowledge of the fundamental theorem (be it new or 
old) by some recent researches connected with my new Rule of Limits, 
considered with reference to the conditions which must be satisfied when one 
of the limits found by the rule comes into actual contact with a root ; 
a contact which I can demonstrate is always possible, as well for the superior 
as for the inferior limits, and with so much the fewer equations (as dis- 
tinguished from inequations) of condition between the coefficients of the 
assumed auxiliary function which the application of the rule of limits 
requires, as there are fewer pairs of imaginary roots in the function whose 
roots are to be limited. 



67] 



the Algorithm of Continued Fractions. 



643 



I may add that the fundamental theorem is an immediate result of the 
representation of the terms of the convergents to a continued fraction under 
the form of determinants. Thus, for example, the determinant 

a, 1 
- 1, 6, 1 
- 1, c, 1 
-I, d,l 
- 1, e, 1 



-1./ 



is obviously decomposable into 



a, 1 


X 


d, 1 


-\,b,l 




- 1, e, 1 


-1, c 




-1./ 



+ 


a, 1 


X 


e, 1 




-1, h 




-1./ 



or into 



or into 



a, 1 
1, h 



c, 1 


+ a X 


- 1, f^, 1 




- 1, e, 1 




-i.y 





d, 1 
- 1, e, 1 
-1,/ 



&, 1 

1, c, 1 

-I, d,\ 

- 1, e, 1 



c, 1 
■\, d,l 
- 1, e, 1 



that is 

{ahcdef) = (a6c) (cZe/) + {ab) (ef) 

= (ab) (cdef) + a (def) 
= a (hcdef) + {cdef). 

Thus the whole of the properties of continued fractions are deduced 
without algebraical calculation from a theorem which itself springs im- 
mediately by inspection from the well-known simple rule for the decom- 
position of determinants. 

If instead of a simple set a triple set of quantities be taken, as 

I n ) ^2 • • ■ H—i I 

, m^ ... vii r ' 
112 ■■■ ni-J 



644 On a Fundamental Rule in Continued Fractions. [67 

which, when i= 1, i=2,i = Z,i = 4, &c. is to be interpreted to mean 



frh'. 



mi, 


k 




7)Vi, 


k 




-Ml, 


ma 


' 


-riu 


— W2, 





k 



-«2> 



&c. respectively, the value of the determinant represented by any such set 
being called Ti, we have in general 

which, when mi and ?i?ij are constant, becomes the characteristic equation 
to an ordinary recurring series. The theorem corresponding to the funda- 
mental theorem for such triple sets will be 



k, 


k . 


•w 1 


'd '2 ••. k—1 


{li+l, k+2 •• 


• w ■ 


m, 


ma . 


. mt+i'+i ■ = 


= ■ TTii, m^ ... mi 


^ X ]m;+i, m,-+2.. 


• m+i'+i ■ 


»ii, 


% . 


. rii+i' . 


?li, Ha ... Mj_i 


^''t+ll '^1+2 .. 


• «£+i' ■ 








'k, k 


..?i-2| pi+2 


..k+i' 








+ ki^i' fThi'nh . 


. . mi_i j- X ■ mj+a 


..mi+i'+i 








Ulj, Wa . 


. . ?lj_„ J Wj+a 


■■rii+i' 



68. 



ON A GENERALIZATION OF THE LAGRANGIAN THEOREM 
OF INTERPOLATION. 

[Philosophical Magazine, VI. (1853), pp. 374 — 376.] 

There is a well-known theorem of Lagrange for determining the form 
of a rational integral function of one variable of the degree m, when its 
values corresponding to wi + 1 values of the variable are assigned. M. Cauchy, 
in his Cours d' Analyse de l'£cole Poly technique, has extended this theorem 
to the case of a rational fraction, of which values corresponding to a sufficient 
number of values of the variable are given ; but the solution of the question 
there given, although of course correct, is unsatisfactory, as it presents the 
numerator and denominator under forms not strictly analogous. 

The theorem of Lagrange, in respect of its subject matter, may be best 
generalized as follows. 

Suppose any number of rational integral functions of x of the several 
degrees mi — 1, mj — 1 . . . mj — 1, say f/j, U^ ■■■ U^, and that the equation 

l,U,+ LU„+...+liUi = 

is known to be satisfied for mj + mj + ... + m; — 1 (say) fi—1 assigned values 
of the system of quantities l^, l^...li, x; there will then be yii — 1 linear 
equations connecting the fi coefficients comprised in Z7i, U^.-.TJi, and 
therefore the ratios of these coefficients, and consequently of the functions 
to one another, may be determined. There is no difficulty in representing, 
by aid of the method of determinants, the result of solving these equations 
whatever be the number of functions ; but for the sake of greater simplicity, 
I shall suppose three only of the several degrees, e — 1, i—1, <»— 1 in sc, 
which I shall call IT, V, W. Now suppose that lU + mV+nW= is known 
to be satisfied for l = lt, m = mt, n = nt, x = xt, t taking all possible values 
from Itoe+i + a—l, say t — 1 ; let the indices 1, 2, 3 ... t — 1 be partitioned 
in every possible way into three groups, containing [when it is the function IT 
which is to be determined] respectively e — l,i and co indices, as 

^1 , 0.2 ... 0e-i ; de, ^e+i • ■ • ^e+i-i '■, ^e+i ■ ■ ■ ^t-i 

41—3 



646 Generalization of a Lagrangian Theorem. [68 

(the terms in any group may be arranged indiiferently in any order, but are 
not to be permuted). Let ^* {p, q,r ... s) denote in general 

(p-5)x(p-r)... x(p-s) 
X (g — r) . . . X (g — s) 



ir, = 2(?) 



X (r — s), 
and write 

1^81 •■ • K-i ; Wee • • • ™fl.+£-i ; %+i • • • "«T-i 

l^i {x, x^^... Xf^_^ fi (a;^,, «e^i . . . «fl^_i) ?* (a;^^ . . . «e^_,)) 

The mark (?) is used to denote (— ) raised to a power whose index is the 
number of exchanges of place whereby the arrangement 1, 2 ... (r — 1) can be 
shifted into the arrangement d^, 6^ ... Or-i- 

In like manner, let 

if* (Xo. . Xi, «<>.") ^^(X. Xa... ...X,,.., .') t^ (X^. ..... «o- .V 



iC* («e„ %, . . . Xe,) ^^ (x, xg^^^ . . . Xg^^__;) ?^ {xg^^ . . . Xg,^ 
and ^3 = S(?)- 



\ . . . Ig,; wiee+i • • • "^ee+^; '^Se+m • • ■ »»er-i 

l^i («9, , «S2 . . . a;^,) ?i («s^i, a;(„+2 • . . x^^) ^* («, aj^^.^j . . . a;^,.,) 
Then, using c to denote any arbitrary constant, we shall have 
U = cK^, V ={-ycK„ W = {-y+icK,- 

and so, in general, the ratios to one another of any number of functions of 
one variable, of which the linear conjunctives for a sufficient number of given 
values of the variable and of the coefficients of conjunction are known to 
vanish, may be expressed in terms of those values. 



EDITOR'S NOTE ON SYLVESTER'S THEOREMS FOR 
DETERMINANTS IN THIS VOLUME. 



In Sylvester's paper No. 37, p. 2-tl above, beside the errors noticed by Sylvester himself, 
pp. 251 and 401 of this volume, the substitution of b, ...b, for a^ ...aj, in line 22 of p. 244 
and the substitution of a. ...a. for 6„ ...6^ in line 8 of p. 245, there is the more fundamental 
error that in formula (2), p. 244, and the formula at the foot of p. 247 the suffixes of the 6's 
should be ki...Tc^ and Zj... ?.,. , and the suffixes of the a's should be Sj...^^ ^■nd <j>^...4>^. It may be 
a convenience to the reader to have at hand another view of Sylvester's three main theorems on 
determinants in this volume (pp. 247, 253, 249). 

1. A matrix of type (m, n) is an object of calculation depending on mn numbers which we 
suppose arranged as a rectangle of m rows and n columns. By the product (a) (i) of two matrices 
(a), (6) of respective types («i, m), (m, «.,) is meant the matrix of type (nj, n,^ which has for its 
(p, g)fh element, that is the }-th element of its^-th row, the number 

flpi 6i5 + ■ - ■ + dp^nfiml ' 
where Op^, b,.g are respectively the (p, r)ih. and (r, q)th elements of (a) and (6). 

If i denote a particular one of the ( ^ ) possible selections of r numbers from 1, 2, ..., n^, say 
ii...iy, and J denote a particular one of the ( ^ ) possible selections of s numbers from 1, 2, ..., n^, 

say jj...Jg, we may pick out from the product matrix (a) (6) a minor matrix of r rows and s 
columns consisting of the elements of this common to the rows ij . . . i^ and the columns j^ . . .j^ ; 
this is clearly given by 

((«) (6) )o- =/«>,! ■■• \m\ / 6iy, ••• KA={a)i{by, 



where {a)( is the matrix of type (r, m) constituted by the i-rows of (a), and (6)^' the matrix of 
type (m, s) constituted by thej-columns of (6). When s=r this matrix is square and, if k denotes 
a selection of)- numbers from 1, 2, ..., m, its determinant is given by 

|((a)(6)),.y| = 2|(a),MI(6y|, 
k 

where l(a)^-^|, which we may denote by |(a)jfc|, denotes the determinant of the minor of (a) 



648 Note. 

formed with its i-rows and its fc-columns, and | {h)^i\ or | {h\j\ denotes the determinant of the 
minor formed with the i-rows and the j-columns of (t), and the summation extends to the 
possible ( j significations of k. Similarly if (a), (6), (c) be matrices of respective types («!, n), 
{?!, m), (m, lij), the minor with the rows i and the columns j of the product matrix (a) (6) (c) of 
type (Hj , n«) is given by 

((a)(6)(<;)),7=((a){6))i(cH = (a)i(6)(cK 
and when s = t, its determinant is 

l(C«)(6){c)),yi = 21({a)(J))^||(cW| 

= 2S|(a)a||(6)Ai||(c)iy|, 
where ft is as before and A denotes a selection of r numbers from 1, 2, ..., n, the summation 
extending to the f ) ( . ) possible significations of Tc, h. 

2. Hence the theorem of Sylvester on the minor determinants of linearly equivalent quadratic 
functions, pp. 244, 247 above. For if by the substitution 

the quadratic form a^iX^"+ ... + 2ai^X2X.2 + ... become b^yi- + ... + 2b^yiy2+ ..., we at once find 

6pg= Z/igp(ag^iJ^,+ ... + agifi!q+...+ag„ix„q), 

so that the matrix of the new form is given by 

(6) = (M)(a)(A<), 
where Op,, /jj^, /i^p, bpq are the (p, g)th elements respectively of the matrices {a), (fi), (/I), (6), 
which are all of type (n, n). Supposing the numbers Hj, n, m, ji, of § 1 aU equal to n, the 
determinant of the {i,j)th minor of order r in (6) is 

it h 
this being the result which in the notation of Sylvester would be written 

h i \% % ■•• <^kr/ \Mi, Mi, •■■ Mir/ Wj ft, ■■■ M,v/ ' 

the first row giving the rows used to form any minor determinant. It will be noticed that the 
columns of the matrix {fj.) which come into consideration are those of the same enumeration as 
the rows and columns of the minor of the matrix (6) which is to be expressed ; this is contrary to 
Sylvester's formula of p. 247 above. 

3. When the product of two square matrices (a), (6), each of type (n, n), is the so-called 
unit matrix, in which every element is zero save those in the diagonal which are each unity, the 
matrices are called inverse ; and we have (a) (6) = 1 = (6) (a). Denoting by a^j the determinant of 
a minor matrix of type (r, r) formed with rows tj ... i^ and columns Jj ...j,. from (a), and by a',y 
the determinant of the complementary matrix of type {n-r, n- r), we have, ii A = \{a)\, yu= ( 1, 
by Laplace's rule for the expansion of a determinant 

a.iii^'n+ .■■ + <>.i^a'jii=A, or, 0, 
according as i=j or i+J. Thus the two matrices of type (^, ii), in which the (i,^')th element of 
the first is a^j, and the {i,.7)th element of the second is -p , are inverse to one another, so that 
we have 

(")(2) = (2)w=(l)(^> = (^'(l) = i' 



Note. 649 

where the bar above the symbol for a matrix indicates the transposed matrix differing from the 
original in having its first, second, ... rows those respectively which were the first, second, ... 
columns of the original. From this equation it is easy to prove that the determinant of the 

matrix (a) is the — th power of A. If (6) be inverse to (a), and ^^j be the determinant of type 

(r, r) formed from (6) as was a,-y from (a), it follows by considering (see § 1) the determinant of 
the product of the matrix {a\ of type (r, n) formed by the f-rows of (a) and the matrix (6)-' of type 
{»i, r) formed by the /-columns of (6), that 

according as i=i or i=t=j ; hence the matrices (a) and {^) are inverse ; and thus, by the above 

Pa- ^ ' 

or in words, any minor determinant of the inverse of a given matrix is equal to the comple- 
mentary determinant formed from the transposed of the original matrix divided by the deter- 
minant of the original matrix. In particular this gives the elements of the inverse matrix 
expressed by minors of the original. 

If (o), (6) be any two matrices of type {n, n) we can form a matrix of type (n, n) by replacing 
the t-th selection of r rows in (a), by the j-th selection of r rows of (b) ; this matrix being called 
{a, b){j and its determinant I a, b |,-y, we have, by Laplace's rule 

\a,b\ij= a'a fti + a'i.^ fe -H . . . -t- a',-^ ftyi ; 
hence the matrix, of type (/x, ii), of which the (;, j)th element is ' — ' " , is given by 

thus, by means of (^) ( ^ J = l, ( -r ) (a) = l, we have 

(^')(iv')=e)*(f)«-. 

and the matrices 

('¥). ('¥). 

are inverse ; this is Sylvester's theorem p. 253 above. 

We remark, using 1 for the unit matrix, the relations, where (c) is of type (n, n), 
l«. l|ij = =''iy. |1. a|ij = «;i. («. 6)jy(c) = (ac, 6c),y, 
of which the last gives, if (6), =(a)-i, be inverse to (a), 

(n-i, l)y(a) = (l, a)y, and hence /3'ij = ^, 
as proved above. 

4. Let n>r>m, and [a] be of type (?^, re). A fixed minor M,,^ of type (??s, m) from (a) 
determines a complementary minor of type (n-m, n-vi), say J/„_,„. From the n-m numbers, 
say p\... Pn-^m ' enumerating the rows of il/„_,„ , make a selection 6^... 9,._„ , and from the numbers, 
say Si..-3n-mi enumerating the columns of Sln-m niake a selection <^j ... 0r-m' tlisn form a 
minor M,. of (a), of type (r, ?■), whose rows are enumerated by those of lf,„ together with 9j...Sr-™> 
and columns by those of M^ together with 0j ... 0,._„j; let the rows and columns of (a) not now 
enumerated be given respectively by S/ ... S'„_j- ^'^<^ 'Pi ■■■ 'P'n-r' 1^' W ^^ inverse to (a). Then 
the determinant of M^ is equal to the determinant formed from (b) with the rows O^' ... 8'„-^ and 
the columns <Pi ... <t>'n-r> multiplied by J. Now suppose ^1,..^^-™ t° become in turn all the 



650 Note. 

I ) possible selections from ^^i ...p„_~, and similarly ^i ... <t>r-m ^^ from q^ ... ?„_„; the 

\r-mj fn-m\ 

determinants M, so obtained form a matrix if of ( ) rows and columns, -which is in fact a 

^ \r-mj 

minor of the previously considered matrix (a). We wish to determine the determinant of this 

matrix if. Now the determinants (8-^ ... 9'„_r. 0i' ■■• 'P'n-r) °^ W' complementary in indices to 

the matrices M^, are minors of the matrix {p^ ■■■Pn-m^ 3i •■■ In-m) °f ('')> ^^^ ^^^ matrix of order 

I ) formed from them has therefore for its determinant A,*', where A, is the determinant of 

\^-'"''J _ / _ _i\ A 

this matrix (p, ... p„ ™, g, ... g„_,,) of (6), and \=( ); hence, as A, = --, where A is the 

\ri rn m> n in— m/ \ i' \ r — m J A 

determinant of the fixed matrix M,„ of (a), the determinant H, of order ( | , of the minora 

m \ I' \r-mj 

M^ of (a), is equal to 

(J)^.^.A^.. 

where ix={ "* ) , (r= ( '" , ) . And this is Sylvester's theorem, p. 249 above. 
\r-nij \r-m-lj 



CAMBRIDGE : PRINTED BT J. AND C. P. CLAY, AT THE UNIVERSITY PRESS. 



10 1305 



